This is for the Vanilla Borderlands 2 and follows the story as close as possible.

If you follow this guide you will not have to use the Gold Chest in Sanctuary. But if you are curious use a key and see what purple items you get.

Basically you only farm for four weapons that will carry you to the ending, this can also be applied in other modes as well. There are some spoilers unfortunately to explain what to do.
Always keep any legendary gear that you favour, and pass it on to your next character, which then lessens the amount of Farming you have to do.

If you are interested on how many missions there are check the wiki link below.
http://borderlands.wikia.com/wiki/Borderlands_2_mission_flow
How you fight the enemies is up to you, this is the basics on getting good weapons/gear to succeed in reaching level 30 to 36 on your first playthrough.

It takes about 18hrs to complete normal mode, but if you are following this guide it will take longer, but it’s not a race, it's about getting there with the most money and good gear/weapons.
Once you get to Claptrap’s place and if you have Digistruct Peak use the Fast travel and go there.

This is an exploit; it is up to you as to whether you want to use it. It does give you extra levels but this becomes negated, by the time you reach your home base Sanctuary.
Do the first mission only, this will give you a skin (Attire cosmetics), you can either sell it or click it and equip at a change machine. Your Character will also attain level 3. Loot every container and if you are lucky you may find a gun or 2, sell them.

If you do not have the Peak then you must collect everything and only keep what is necessary and sell the rest at all times. The only exception is if your backpack is full, you find something better than you have, and you are not near a vendor to sell items. Doing this sets you up with money to buy items if you need to, from vendor machines. Fast travel back to Claptrap and continue on.

Once you reach level 5 you get your first skill point for your action skill, again there are plenty of guides on where to assign them, it's up to you whether you put them in survival or damage skills.
Your first Mini Boss is Knuckle Dragger, he drops the legendary Hornet pistol (Corrosive Elemental weapon) if you are lucky. If he does not drop it, continue on, hand in your quest and get a shotgun.

Go through the door, save and quit, select character and continue,turn around re-enter the area. This is your first Farm for the Hornet. Kill the boss if no hornet, save and quit rinse and repeat till you get it. Time consuming but well worth the effort, as the pistol will last until you acquire the Teapot from Tiny Tina later on.

Once you have the pistol, continue on, loot everything that you can, and do all the side quests given to you.

From this time on look for the Borderlands symbols, some are well hidden, hint, explore everywhere, there are guides on where they are in the forums. Finding them gives you Badass points that assist in buffing your character.

Check every vendor from now on, as some of the gear can be better than what you have, look at the stats and compare if it’s worth buying. Gear is colour coded, white, green, blue, purple and orange, with white usually having the worst stats, but not always.

You will also acquire your first shield, again look at the stats and upgrade accordingly. A white can be better than a green so choose carefully. There are 3 types that I recommend that you equip when necessary, Turtle (survival), Absorb (ammo) and Spike Shields (Varkid’s etc). Again it is up to you whether you choose these or other types.

Class mods, you may not acquire one until you reach Sanctuary from the vending machines, if you do, equip and use it, and check the vendors for a better one, I recommend going for healing but it’s up to you.

You may also find Eridium from now on, you need it to upgrade your backpack carrying capacity and ammo/grenade capacity which can be done in Sanctuary your home base once you reach it. Upgrade your backpack carrying capacity first then go for pistol, sniper and grenade upgrades.
Try to keep a balance between backpack and ammo upgrades.

With the mission Bad Hair Day, I suggest getting the sniper from Hammerlock. Another mission Symbiosis is to defeat the Mini Boss Midgemong who drops the legendary Kerblaster, nice if you can get it, but not essential for solo. Hint check the telescopes out in Southern Shelf.

Your next Mini Boss is Boom and Bewm both of them can drop a legendary grenade the Bonus Package, it is not essential but great if it drops. You can farm for this if you want as this grenade does pack a punch, your decision. I recommend using the hornet for this fight when you enter the area veer to the left for this fight, but it’s up to you. Hint there is a hidden container in here, check the lift.

Your next Mini Boss is Capt Flynt and is no pushover. When you drop down into the area there is an open ended container to your left, use this as cover, use a sniper for the minions and again use the hornet for Flynt and 3 more minions, he drops the legendary pistol the Thunderball Fists, great if it drops but not essential. Continue on, when you reach Sanctuary you should be around level 9.

From this point on, there are too many missions to go into detail, so I will be selective on what missions you need to farm for weapons, others I will skip that do drop legendary items but are not essential for Solo, and some recommendations on how to get past certain points in your adventure.

Once you reach level 10/11 check the vendors for a rocket launcher and buy one, be mindful of the stats. Do not get the 2 shot variant, load times are bad. Also keep an eye out for tesla grenades lvl10 or above.

You get quite a few missions in Sanctuary grab them all, and do the mission the Name Game first once you have completed other missions within Sanctuary. The reason is you can then farm your next Mini Boss Savage Lee in the marrow fields (He can spawn in two places, near the fast travel and the Marrow fields giving you an extra attempt) for the Legendary pistol the Harold which comes in different variants, but any of them will do the job. This does involve some driving but this pistol is well worth the effort.

Your next Mini Boss is Doc Mercy in Three Horns Valley who can be farmed for the legendary Infinity Pistol, though it’s not necessary for Solo. The pistol has the advantage of saving ammo so it’s up to you if you want to farm for it. There are guides on the quickest way to do this.

Continue doing all the missions until you have to do False Idols in Frostburn Canyon. You have to kill a Mini Boss Scorch, use your Harold on him, he can be a handful, but his minions are more of a problem than him. He drops the legendary weapon the Hellfire SMG. If it does not drop hand in the mission, grab the next mission save and quit. Continue, and farm Scorch (Run past all the enemies don’t fight them, save your ammo and time for Scorch) using the save quit rinse and repeat method. In an area nearby there is also a blue SMG the Lascaux which can be acquired if you want it.

Your next major hurdle is the Bloodshot Stronghold. When you enter this area there are two major fights in this section, Mad Mike who shoots rockets at you, so watch out for him as he can kill you very easily and then a constructor (W4R-D3N) at the end.

I will concentrate on the constructor. When you get to him, the fastest way is to use tesla grenades, as many as you have, and hit him with a rocket launcher (A five shot or better if you can get one). When you enter the area turn left go behind the container, turn right at the end and start unloading your grenades and fire your rocket launcher, reload, then run back around and hit him from the front if you need to. There are other ways to do this, but this way always works for me. You need to eliminate the constructor, otherwise you will have to fight it in another area in the dust if it escapes.

Your next adventure is Tundra Express by now you should be around level 14/15 or slightly higher. There is a mission that Hammerlock gives you called Mighty Morphin in Sanctuary ignore it and do it later. Also while there, buy a spike shield.

In Tundra for the first time there is a snowman, shoot it and get a higher level weapon, sell or keep it. Equip a spike shield so you can get badass points from the varkids. Use your hellfire for them. Once you get the mission, You are Cordially invited Party Prep you have to fight a Mini Boss Madame Von Bartlesby. I will leave you to figure out how to kill it (Hint use the Harold, if your having trouble jump off the plateau use a fire sniper and keep your distance from her) . The legendary Babymaker SMG can drop but not essential. You may also find a relic, recommend a cooldown one for action skill.

You need to complete all the Tiny Tina missions in order to get the pistol the Teapot a blue reward to fight your next Mini Boss Wilhelm. You can use your Harold but you might as well use the Teapot with a couple of tesla grenades. Wilhelm drops 2 legendary items neither of them are essential, though the Logans gun is handy to have.

Once Sanctuary has flown into the air you have to get to the Fridge, whilst in there go into the slaughter Dome area and upgrade as much as possible, by buying gear in the vendors by save quitting rinse and repeat. Before you continue on make sure you have a shock sniper/weapon.

Continue on and ignore the Crystalisks, just get through to the next area. You will have to come back to the area later on with other missions.

Once in the Highlands Outwash you have to fight your way through various enemies and you come to an area where there is a drawbridge and if you are not quick enough to kill a loader, the bridge becomes inaccessible. After you finally figure out how to get across, you fight a constructor and some minions, make sure you lower the drawbridge if it is up.

You now have to fight a Gluttonous Thresher, to retrieve a beacon which it swallows, walk up to where the beacon is that you have to get, at the first sign of it, turn around run back across the bridge and hide behind the control system to the left of the bridge. Use your shock weapon to lower its shield and lay into it with any other type of weapon. Keep switching if the shield comes back. I suggest this method as it’s the easiest option to beat it.

Once back in Sanctuary after an epic fight to get there, grab the mission The good the bad the Mordecai. Once the mission is completed, drive back to Ellie's garage, save quit, and farm the two characters Mobley and Gettle. What you want is the legendary sniper the Lyuda and Gettle drops it. To farm, it involves some driving but is fairly quick to do. If your not sure how to farm these two look at some vids on the best way to do it. Try to get a corrosive sniper if you can as it will help with other missions later on.

The level of the weapon ranges from 18 to 22. Later on you can return and farm them for a level 28/29 as they level up, suggest going back after you get the mission Data Mining or after completing it.

You can farm the Sheriff, for the sheriff’s badge in Lynchwood in earlier missions but it is not essential to have it, though it does benefit Salvadore the most.

Loot midgets in the Wildlife Preservation, they can be farmed for legendary gear by getting the Mission Doctors Orders from Tannis, and by not picking up any of the recorders in this area. There are guides on how to do this the fastest way.

Wildlife Preservation at the end you have to fight another Mini Boss and to save heartache I won’t name it, the boss goes through 4 stages, fire, shock, corrosion and slag. Either use a non-elemental powerful weapon or elementals that are not the same as the boss is.
Do not start shooting until all the commentary has finished. You need to down it as quick as possible usually by the third stage if possible, otherwise it becomes a slog, and skags start appearing to further complicate things.

One other piece of gear you can farm for is the Shield, The Bee, again not essential but handy to have when in a vehicle and you have it equipped. You have to Farm Hunter Hellquist in the Arid Nexus a quick short run once you have killed him and handed in the mission.

There is a Mini Boss in the Arid Nexus Boneyard, Saturn, which can be quite difficult, you need to use corrosive weapons against it and use cover otherwise you will not last long. Suggest using the main building in this area, or the fence before you enter the area where the Boss is. If you have homing grenades use them and blast away, your only issue could be ammo, so plan for this if you have to get more. For Zer0 with the skill Bore, this boss is a pushover.

From here on in the ultimate target is the Warrior, use the Lyuda and aim at his mouth, again there are guides on how to do this. You can farm the Warrior by save quitting and selecting normal mode on the menu, go back kill, save/quit rinse and repeat. It’s a good source for Eridium and money plus other gear. If you are very lucky a legendary shotgun the Conference Call may drop and can be used by other characters to fight the Warrior with ease.

Once you are ready to move on, suggest level 35/36, save quit select TVHM and start all over again. In TVHM, be mindful that some Mini Bosses do not level up, until you reach level 50 and have defeated the Warrior, where as in UVHM they do and you need to Farm every 4/5 levels.

By now you should have a fair idea of what to expect in the vanilla game and can plan accordingly for other modes. I have not mentioned the DLC’s, it’s up to you when you want to complete them if you have them.

Please rate this guide up if you find it helpful.

Happy shooting and looting !!