Role: Defensive Ranged combatant / Buffer
Soldiers are your base line troopers, and as such are one of only 2 classes in the game costing a mere 400 SP to deploy. They are also one of the 2 classes to get the powerful overwatch ability, and the second best buffing class in the game.
So.. not bad for a "base line trooper", right?

Skills:
Burst fire: ranged attack with small cone spread at target. 3 heat.
Overwatch: shoot anything that enters the specified area. drains all AP and MP when used. 6 heat + 6 heat/shot.
Calling Shots: The soldier's weakest buff, boosts Ranged Accuracy and Critical Chance for the team. 2 AP 15 heat
Concussive Grenade: Throws a grenade for moderate damage + debuffs on enemies hit. 1 AP 6 heat
Shredding Fire: this limited-range attack is similar to burst fire but has added penetration to crack through heavy targets. 2 AP 6 heat
Lock Down: Adds Armor, Parry dice, and at higher ranks Resistance. Ends movement for the turn. Self only. 1 AP 15 heat
Defensive Stand: This is the second best soldier buff, improving armor, Auto-Block, and adding parry and dodge dice to all those effected. Usable in overheat. 2 AP, 15 heat
Rallying Charge: Best buff, possibly in the game. +damage and +MP. 1 AP, 6 heat

Weaponry and Gear:
The soldier's weapon options consist of 2 types: Plasma and Rifle.
Plasma weapons are shorter range, have great penetration, added plasma type damage, but lower standard damage than comparable rifles.
Rifles have solid range, but lack stopping power, often requiring multiple hits to down even a skitterling.

I always get a soldier to Rallying Charge 4, because in the few missions where you're escorting non-templars they move quite slowly. This talent fixes that and is an integral part of getting your turn based bonus on those missions.
I often start stacking up Defensive stand after that because it can greatly reduce the chance of a teammate taking damage- particularly useful for the captain himself.
While I use Overwatch and burst fire the most, I don't generally rank those talents because my hit chances are good enough and the damage added from more ranks is not reliable.
I generally prefer Rifles for their range but against high armor enemies I will sometimes bring in a plasma soldier.

Role: Rapid Recon / Sniper
Scouts have the potential to be the highest single target damage dealer on your team by using the headshot talent, as well as being by a great distance the fastest templars on your team through the overdrive talent. They suffer severe deficiencies in their ability (or lack thereof) to deal with multiple targets and their frailty (1-2 hits and dead). They can also be very difficult to learn how to use due to the stealth mechanic being tricky to learn the proper use of.

Piercing Fire basic shot the reduces enemy armor and deflection, allowing other teammates to deal greater damage to the target. 6 heat
Sensorkit Scan reveals an area. Cannot be used when overheated. Shared skill between engineers and scouts. 6 heat unless max ranked (then only 3 heat)
Crippling shot fire a shot reducing dodge and MP of the target. Personally, this is my most used scout skill. 6 heat.
Headshot fire a shot with greatly increased critical chance. Depletes player movement when fired. 6 heat.
Field of Focus self only buff -MP +damage and +Critical chance. 1 AP 9 heat. Cannot be used while overheated.
Overdrive: boosts movement and stealth. self only.. 0 AP 6 Heat
Null Field: boosts stealth (more than overdrive does). self only. 2 AP 15 heat. Cannot be used while overheated.

Weapon options include sniper rifles and needle rifles. Both offer excellent range. The sniper rifle is great for dealing massive damage to a single target, where the needle rifle is used to deal more stable non critical damage but it's spread over time. Most needle rifle builds make use of crippling fire to slow the enemy so that they will not make it into range of your team before the bio poison has finished doing it's work.
Sniper rifles benefit far more from high critical chance than needle rifles do and scout's highest damage move is headshot, so many players prefer sniper rifles over needlers when building a scout for damage dealing.

I've read a lot of great things about headshot scouts, but I've never used one long term. I favor a build of overdrive 3 (past rank 3 it stops giving another +1 MP every rank) Crippling fire 3 (-3 MP nearly immobilizes anything but skitterlings) and the rest into piercing fire (more -armor and -deflection always helps and this becomes the primary attack when just shooting at something really tough)

Role: Tac point capture / general assist
The capture talent, unique to engineers, is needed for many different missions you'll be going on. Engineers can build turrets, repair turrets or tac points, buff allies, and target allies with heat removal talents.
Sounds amazing, doesn't it? Here's the catch: everything they do is pretty horrible unless rather well ranked, meaning most engineers will only have at most 1 skill past the absolutely critical capture talent. Also engineers aren't that great in a firefight no matter how many points you put into direct shot (hint: don't waste points there.)
The sheer variety of things competing for points on engineers is what made soldiers take second best buffer instead of it going to the engineer.

Direct shot: Fires your primary weapon at an enemy for 3 AP.
Sensorkit Scan reveals an area. Cannot be used when overheated. Shared skill between engineers and scouts. 2 AP 6 heat (3 heat @ rank 10)
Capture: used to take tac points. only skill I'm aware of where the AP cost changes based on skill rank, going from 3 AP at ranks 1-5 to 2 AP (meaning do it twice in a turn) at rank 6. This makes Capture 6 the most critical engineer skill that nearly every engineer should have. 15 heat.
Repair: heals a sentry turret or heals (a lot less) to a tac point. Limited uses. 3 AP 9-15 heat (based on rank)
Power Field: the only tactics based buff of the engineer class, it grants dodge, parry, deflection, and armor to your team. 2 AP, 15 heat.
Fortify Position: grants critical chance, deflection, auto-block, and Armor to your teammates. Better Critical chance bonuses per rank than the soldier's Calling Shots. Reduces MP of buffed soldiers for the duration of the skill. 3 AP, 15 heat, usable in overheat.
Sentry Turret: A gun turret with burst fire and Overwatch. 3 AP 30 heat
Scorch Order: debuffing order, - dodge and -flame resistance. 1 AP 6 heat
Turret Stance: Immobilizes templar and gives +damage, penetration, deflection, and critical. 1 AP 9 heat, 3 turns
Landmine: deploys a landmine. limited use, 3 AP 12 heat.

Normal Armor powers:
Heat Sink: removes a large amount of heat from a nearby templar. AP 2 heat (15-22, higher ranks cost more heat)

Pyro Armor powers:
Manual Vent: bleeds minor amount of heat from other templars and grants -MP + Critical and +ACC to them. The -MP helps reduce their heat even more, and between this and fortify position you can give your team a massive critical chance boost. Duration not listed but I believe is 3 turns.
Pyro Turret: A shorter range turret based on flamethrower tech. Has limited uses of a napalm ability plus a basic attack with good spread. 3 AP 30 heat

This is the first class to have 3 sections of skills; this is because some skills are locked to certain loadouts.
I tend to favor non pyro engineers but that's largely because I'm used to working with them due to not prioritizing the unlocks needed for pyros. Also, a Pyro engineer cannot deal with a spore pod like a hydra can (I've tried). The shorter range of hand flamers as opposed to pistols also puts a pyro engineer a little closer to the front. That being said, Pyro engineers hit harder than non pyros.
While the Pyro turret is restricted to Flame tank armor, the sentry turret is NOT restricted to non flame tank armor. This means that you could have four turrets out from one engineer, although you'd probably have to sacrifice for it on his capture talent and that engineer will have crazy heat issues.

Non Pyro engineers can hit suprisingly hard with some of the later pistols but the lack of area attacks and weaker than typical defenses generally reserve them to cleanup detail on wounded enemies.
Either type of engineer can equip a reactor, which can boost MP. As engineers often are started 7 spaces away from tac points you're expected to take in missions, I advise universally equipping them for 7 MP. Because this can take a while to reach in the tech tree, The cyclone reactor is the only relic I advise unlocking early in the entire game. The extra heat allowance from it can be an immense boon on higher difficulties.

Role: Offensive Ranged combatant

The Hydra is your fire specialist, wielding a promethium powered flamethrower. They suffer suprisingly few heat issues and deal often suprising amounts of damage, as the majority of their damage output is fire typed instead of normal type, thus making them great for cutting through most armor. While they do struggle with Magma Carapaces, the things that resist Hydras are generally vulnerable to other offensive specialists. This makes the hydra a top tier assisting asset to an offensive group. Their ability to equip reactors also makes them one of the few classes that can have a base speed of 7 MP.

Skills:
Hellfire: this primary attack skill has the same area of effect scaling as the Neptune's Full Auto, meaning at rank 5 it hits a large area. Heat 6
Napalm Charge: This skill lays down a huge pool of fire on the ground, damaging anything foolish enough to enter it. It has the unique property of at rank 8 being the largest AOE in the quadrant. Heat 15 AP 3
Incinerate: close range high damage single target fire attack. Generally not enough more damage compared to hellstorm for my personal tastes, but has been absolutely devastating when I have used it. Heat 15
Heat Waves: This short range debuffing attack hits the same area as a rank 5 hellstorm at rank 1 and is the only debuffing attack in the game to give -damage to victims. 9 heat
Inferno Seal: A team buff booting Armor, Deflection, and Fire Resistance. 9 heat. Can be used while overheated.
Redline Reactor: temporarily increases a Hydra's Damage output and Heat Cap. A hydra with this at high ranks can basically ignore the heat mechanic, but if it isn't renewed before it runs out that hydra may be in for a very long shutdown.(max rank it gives +130 heat cap, so that hydra barely registering in overheat range could suddenly be past their shutdown limit without it) 1 AP 6 heat.
Napalm Lance: Similar to Napalm Charge, but with large bonuses to critical chance and limited range and area (this never hits more tiles than a level 1 napalm charge). When I'm considering this skill against napalm charge for my talent points I must confess, this skill has never won. Because both skills lay patches of fire on the ground, the damage of which is based off the talent level which laid down the flames.. yeah. Heat 15, AP 3

The reasons I classify Hydras as an offensive class despite "Lake Promethium" being a well known defensive measure is because they have a tendency not to get bogged down and lack the ability to hold areas for nigh-indefinite durations (they'll eventually need to resupply their promethium or use a medikit). Those facts stated, they can do immense amounts of damage. If using a hydra to lead an offensive charge I'll grant Napalm Lance may be more useful than Napalm Charge, due to the limited area of lance meaning less fire for the rest of the team to run through. Inferno seal can be really useful here as well for protecting your teammates from your own fires, though I generally don't use it myself.

Role: Defensive Melee Combatant / Buffer / Healer
The paladin is a skill challenged class like the engineer. The difference is that with healing and a shield, the paladin can at least serve as a front line bulwark if all his skills are in team assistance abilities. Paladins, like berserkers, suffer moderate to severe heat issues.
Note that all Paladin group buffs can be used while overheated!

Strike: basic melee attack.
Battlefield Medic: heals a target for a large amount of health. uses 2 AP and 15 heat, touch range only.
Battlefield Enhancers: Heals a minor amount of health to all those affected and applies the strongest + damage buff in the game to all of them.Also gives some dodge. Uses 1 AP and 15 heat
Warding Fire: Despite the high heat cost (30) this is arguably the most powerful buff in the game. At max rank, this will put all templars affected by it at their cap for Auto Block. It also gives some dodge. Uses 2 AP and 30 heat
Inspiration: The paladin's "weak" buff that gives parry, melee accuracy, and damage boosts to the team. 1 AP and 15 heat. The damage boost of this talent is about 1/3 of the boost from battlefield enhancers.
Devastating Strikes: A melee attack that gives -ACC and -dodge to the target. 6 heat
Shalun's Wrath: melee attack that gives the paladin a +damage, + critical chance, and at higher ranks +ACC self buff. 6 heat
War Machine: Moderate self heal, -MP and +max heat, deflection, and auto block self buff. Limited Uses. 1 AP 6 heat

Shield Only Skills:
Shieldbearer: boosts parry and auto block for the paladin. Limited Uses. 1 AP 6 heat

Hammer Only Skills:
Righteous Embrace: gives a buff that while active lets you get other buffs depending on how many targets you can take out in one swing. 1 AP 12 heat
Staggering Blows: swing hammer at a group of targets with + plasma damage and +penetration. limited uses per mission but the amount is rather generous. 9 heat

Sword and board vs. big hammer...
Big hammer is functionally a paladin playing at being a berserk. That being said, they play that game rather well. Big hammer is reliant on using righteous embrace and following quickly after it with staggering blows to rack up full benefits. If you can get that active with ultra, consider using prolonged deployment on this paladin.
Sword and board can reach the autoblock cap (59%+focus) passively with the right gear. It leaves a lot to be desired offensively (good autoblock swords tend towards poor penetration) but makes for a potent enough bulwark.
Paladins are the only class with 3 different recharge ordinances: one for battlefield medic and battlefield enhancers, one for shieldbearer, inspiration, and warding fire, and the final one for war machine, righteous embrace, and staggering blows.
This means that a "hold the line" build using war machine for smaller self heals, battlefield medic for larger self heals, and shieldbearer to block needs all 3 to recharge. While I've used that build for my testing on relliar's shield, I generally advise either
1) go full hammer, biggest hammer you can carry, make sure to respec talents to reclaim the point that started in shield block, and focus on war machine, righteous embrace, and staggering blows
-or-
2) get as close as you can to passive autoblock capping, get just enough shieldbearer to make up the difference, then go medic with battlefield enhancers. If you can reach cap without shieldbearer, do so
there is another way:
3) build for warding fire/ battlefield enhancers, designed so that any difference between your passive autoblock and the cap is made up for by your ranks in warding fire. This paladin's job isn't to fight, it's to buff. Beware the massive heat of warding fire if you use this build.

Role: Defensive ranged combatant

The neptune is known for it's "sheer crushing power" (quoting the developers there!) on the battlefield, and is a master of area attacks. They have the highest penetration non melee weapons in the team and the overwatch talent. They also are more prone to heat problems than most primary combatants, and have MP reducing talents.

Skills:
Full Auto: This talent shouldn't be used until it's rank 5 and hits a bigger crowd. deadly burst 1 hits the same area for 9 less heat. After rank 5, the area advantage of full auto becomes impressive. 15 heat
Overwatch Burst: Like overwatch, 'cept you're firing a machine gun. Generates 5 heat on activation and per shot fired.
Deadly Burst: This is the only neptune talent that does NOT end your movement for the turn. I advise a minimum of 1 rank on every neptune. 6 heat
Concentrated Fire: The only neptune talent that doesn't threaten the target's friends & neighbors. Suffers a shortened range. 9 heat.
Stabilize Platform: Boosts Ranged accuracy, Critical, and Damage for 4 turns. Self Only. 1 AP, 15 heat.
Full Vent: a heat dropping talent that debuffs your defense for a few turns, along with debuffing your movement. That being said, the -MP helps it cure more heat over time. (heat reduction starts at 25, gets better with ranks) 1 AP, 0 heat
Suppressing Fire: The mother of all Neptune attacks, this hits a large group of enemies and instills the potent -MP debuff to the enemy. Becomes an impressive 13 tile spread at rank 5. 15 heat.

Weaponry and Gear:
Neptunes don't really have alternate weapon options right now, but there is talk of them gaining a grenade launcher skill tree.
Generally neptune's weapons are the same range as soldier rifles, with 10-20 more damage and 20% better penetration.

Neptunes can struggle with heat, especially without deadly burst. As all their attacks but deadly burst remove all their MP, heat reduction due to unused MP will not occur with any other attack, even if standing still and firing downrange.
Also if going past rank 2 in full vent, make it a point to skip rank 3. the debuff is worse than rank 2's, but heat reduction remains the same.
Full Auto and Suppressing Fire both get bigger areas at rank 5. I advise getting both to 5 after getting deadly burst 1 and full vent 1.

Role: Offensive Melee Combatant specializing in counterattacks and armor penetration
That's right, ONE ROLE. The berserker is the least ranged, least flexible class in the game. The hardest to properly use. It's a rare situation that plays to their very specific skillset, but they are devastatingly powerful in that situation. Full disclosure: my survival rate for berserkers would probably get me thrown out an airlock.

Talents:
Axe Swing: basic attack, 3 heat
Unbalancing blows: -MP, -ACC, and -AP debuffing melee attack. 3 heat.
Battle Lock: +armor, -MP, + ACC, +damage self buff. Limited uses. 1 AP 15 heat
Assault Grenade: grenade with extra large area and more uses at high rank than other grenades. 1 AP 6 heat.
Uncapped Fury: minor self heal, + damage, +crit, -parry. limited uses. 1 AP 12 heat
Kill Strike: Attack with bonus penetration and plasma damage. Damage buffed in recent patch made skill more competitive. Limited uses. 9 heat
Berserk Slash: area attack unique in it's lack of an ACC bonus to hit. + damage, pen, and crit though. 9 heat

Shield only:
Warding Shield: + parry, armor, and auto block. 1 AP 15 heat
Indomitable Defense: + large amount of counterattack%, unlimited counterattacks, + damage. 0 AP 15 heat

Dual axes only:
Wrathful Defense + deflection, + Auto Block, and unlimited counterattacks; dual axes only, lasts 2 turns. 1 AP 9 heat
Hacking Blows: self buff, + ACC, +pen, +Damage, -Parry. 1 AP 9 heat.

A dual axes berserker is the most dangerous melee templar around, period. The recent inclusion of an ordinance to resupply their self buffs means these guys can really wreck xenos for days now.

Shield berserkers can generally outlast a dual axe berserker, but their damage is lower resulting in more enemy attacks coming at their comparable defenses. Further their need to throw more attacks to kill targets and use multiple buffs to defend and counter instead of an all-in-one buff leaves them more vulnerable to heat issues, which are a defining problem of this templar class.

The grenadier berserker build deserves a special mention here: it's where a berserker gets assault grenade ranked really high and uses the roavin detonator relic to make them automatically penetrate. It's 6 shots/mission of house cleaning awesome, but the berserk will generally be suboptimal in combat performance aside from splattering crowds with grenades.

The captain is the only member of your team who is not optional and will be in every mission. if he dies, you lose.
He is the only team member capable of giving +critical vulnerability to enemies, through the skills Fire Commander and Denounce. As it takes a scout of either rediculous level or outside buffs to have a 100% critical attack, this can be a very useful tool in your arsenal. With soldier's calling shots max rank, engineer's Manual Vent max rank, and engineer's fortify position max ranks plus fire command max rank on a target, there is a 78% chance to critical that target. accounting for 16 tactics on shooters, that's 94%.This means so long as each shooter can acquire 6% critical from gear or skills (every class can do it), this skill can allow your entire team to gaurantee critical hits. Only the Captain can do that.
Of course,
I'll be dealing with captain abilities in a few groups of what I consider "related" skills.

Melee Skills:
These have + penetration with the exception of Denounce, which is amazing for other reasons.
Slice: 3 heat basic attack. used in counterattacks.
Denounce: one of the best debuffing attacks in the game actually, gives - armor and + crit vulnerability, with a dash of - parry for spice. cap leads with this, sniper headshots, enemy explodes. 9 heat
Bladeweave: Melee range burst style area attack. 9 heat

Pistol Skills:
As these cost you the ability to wield a shield, I would advise you not to enter melee if ranking into any of these.
Precison Shot: Basic pistol attack, 3 heat
Honing: + Damage and + critical self only buffing attack.12 heat, lasts 3 turns.
Arc Fire: Cone shaped pistol area attack, boosted critical chance. 6 heat

Buffing Skills:
Commander's Confidence: Team buff that improves ALL accuracy and dodge. can be used at a greater range than most team buffs (although that'd mean the captain misses out, so of dubious utility!) 1AP 6 heat
Pressing Need: +1 MP (2 at rank 4, 3 at rank 8+) and +dodge to a single adjacent ally. 1AP 6 heat
Warrior's Wrath: +MP, +ACC, + damage, and +Critical chance at high ranks for the captain. 2 AP, 6 heat.

Command Skills:
These both have excellent ranges at which they can be applied to enemies and max targets per tactics.
Fire Commander: -dodge and + Critical Vulnerability to a group of enemies. 1 AP 9 heat
Discerning Eye: - parry and - deflection to a group of enemies. 1 AP 9 heat

"Oh right, I forgot the grenades!" skill:
Frag Grenade: basic grenade attack with good pen and decent damage. 3 uses at rank 1, 7 uses at rank 10. You got to try it out in the tutorial, you know what's up with this. :) 1 AP 6 heat

Shield vs. pistol amounts to you getting to choose if you can avoid hits somewhat or can fight from short range without eating your grenade supply. While it is possible to use a pistol and still be at the autoblock cap, that relies on outside buffs. (I use a soldier's defensive stand for this)

Fire commander and Discerning eye are only good against tougher enemies because if you use them against any/every enemy, the +9 heat will roast you. That being said, there's a lot of tougher enemies in the late game and you can hit a full 3x3 spread with these skills at rank 6 or above. I favor Fire Commander, with my crit favoring builds my overwatchers have better than 2/3 odds of critting each shot. (I think 28% of that chance is from the captain, rest is gear, etc.)

The commander's buffing skills are generally sub-optimal (that's just north of being called flat out bad) although getting a single point into pressing need is advised for melee captains. (1 extra MP for 1 AP gets you in range to swing a suprising number of times. WW costs 2 AP and thus reduces the number of attacks you can output that round by 1.)

Grenade tree on the captain is something I've never been willing to risk trying past rank 3. Limited uses on an attack scares me away because if I run into trouble, recovery becomes difficult without ammo. Usually you run into trouble a few turns after your grenades ran out with grenadiers.

Honing is a hot mess: this costs 12 heat for a 3 turn self only buff, if it hits. The sheer heat of this skill makes me iffy on it, especially since a pistol captain is still firing a PISTOL, he'll generate a bit more heat than most templars by virtue of having to move into range of targets more frequently. My view is biased by my use of "command" category skills, which already are higher heat than most.
Honing is actually the only captain move costing more than 9 heat.

While above I went into the heat issues of honing, I wanted to touch on Denounce. Denounce is, in my opinion, the single most powerful debuffing move in the game for tough enemies. +critical vulnerability, -armor is a mean combo. The catch being that your captain, guy who if he dies you lose, is now next to that tough thing.

Bladeweave is great when it's good, but it's application should be rare because the captain must not die, therefore you shouldn't be putting him into close range situations where enemy concentrations are high enough to enable frequent use of this skill. That being said, if he finds himself there this might save his life. I advise 1 rank for melee captains. Any more and you'll get tempted to try something dumb, any less and you've lost the option.

Pro grenade tip: rank 6 fire commander makes up for grenades having poor to hit at low ranks. The extra chance to critical is just bonus.