A lot of these quests will involve summoning npcs or performing a certain action to advance, some of these actions consist of summoning npcs. Summoning an npc MUST be done before defeating the area boss. If you don't do this, you can lock yourself out of certain questlines or from receiving items for the entire playthrough. Some of these items are weapons and armor, but others are spells, rings, and gestures, which you will need for some achievements. For this reason, I have structured the guide's sections around the bosses you shouldn't kill until you complete the steps. Some of this is just efficiency, so warnings will also be there separately for each event. Also, make sure you're exhausting all the npcs you meet's dialogues. This means talking to them until they start repeating themselves. If you're in a rush you can press the talk button to skip dialogue, but I recommend listening to it to get a better idea of the characters also I'd say you're doing yourself a disservice since the voice acting in this game is phenomenal. This is an essential mechanic in dark souls to advance character questlines. Also usually when someone asks you a question, pick the response that agrees with them but if in doubt check a wiki because you can mess these up very easily. You can also back up your save file in case something goes wrong, which you can find in Users/your user name/AppData/Roaming/Dark Souls III/DSsomenumbers.sl2. AppData is a hidden folder, so make sure to click show hidden folders if you don't see it. If you have trouble, there are plenty of guides on youtube that can help you.

Do note that it is impossible to complete Sirris' quest line AND get the master of expression emote in ng, but it is relatively easy to get the missing gesture in ng+ with only killing gundyr, vordt, and sages (which shouldn't be as hard the second time in ng+), so I recommend finishing sirris' questline in ng first, just make sure not to forget any late game gestures.

This guide's information is based on a post by fextralife user Dapiek_Absaroka, the fextralife wiki, and stuff I discovered during my playthrough. I hope it helps to have the information all in one place and easy accessible from the shift+tab game menu.

• Meet Hawkwood at Firelink Shrine.
  
• Free Greirat from the cell in Lothric High Wall. You can find the key in the room guarded by the spear knight which leads to another room swarming with enemies (dogs and axe and halberd guys). It also has an estus and titanite shard, so although it's a pain to do so, try to clear out the area or do a suicide run to get all the items there.
  
• After getting the second bonfire in the High Wall, Leonhard will appear in Firelink.
  
• Recruit both Yoel (near the cliff of the starting area) and Cornyx (cage near firebombers) from the Undead Settlement.
  
• Deliver Greirat Loretta's bone from the Undead Settlement. After which, send him off to plunder the Settlement. He will come back after defeating any boss with updated wares, so make sure you're checking if he's now selling anything you want when he comes back.
  
• Meet Siegward in Undead Settlement and beat the demon there together. You can die in this step, but don't let siegward die because this will abort his questline. If you think he'll die let yourself die and it'll reset the encounter next time. Once you kill it, share a toast and a nap with the onion knight. You can also collect a pale tongue here if your invasion attempts were unsuccessful. When you come back also go to the top of the tower to make peace with the giant. He drops a ring when killed, but you'll want to keep him alive for now, you'll understand later why. If you really want the ring, come back to kill him once you've finished the game.
  
• Climb in the sketchy guy's cage to meet Hodrick in the Pit of Hollows and gain access to the Mound Makers covenant. Hodrick can also invade you in the Undead Settlement and will drop a Vertebra Shackle. Again, the invasion will not be possible after beating the area boss.
  
  
• After getting the Pale Tongue from an invasion or the Siegward area, Leonhard will provide the key to the dungeon in the High Wall. Beat the Darkwraith there to get a Red Eye Orb (infinite invasions). Talk to him again after to get the applause emote.
  
• After allying with the giant, collect the mortician's ashes by the tree. This will unlock the key to free Irina of Carim and get Eygon to ally with you. DO NOT give Irina either the Deep or Londor Tomes.

• Meet Anri and Horace at Road of Sacrifice. Get Blue Sentinel's covenant from Horace.
  
• Yellowfinger Heysel will invade in Road of Sacrifice, beat her for items. This is also necessary to unlock an optional miniquest if you're following the Rosaria covenant quest.
  
• Recruit Orbeck at Road of Sacrifice; this requires a minimum of 10 INT but take note that you can put off recruiting Orbeck until very late. Also, by using the Scholar's Ring you can circumvent actually investing anything into INT. I recommend recruiting Orbeck later, because his presence will cause a stop in Yuria's questline until he dies, which will happen naturally throughout his questline, but your only reward for this is a weapon, which will probably be useless as you already have an upgraded weapon you're used to at this stage of the game.
  
• After reaching Road of Sacrifices, meet Sirris in Firelink.

• Meet up with Siegward at Cathedral of the Deep. He is in the well and will "appear" after you unlock the third doorway leading to the Cathedral's central bonfire. You have to do this before the next step/Patches spawns.
  
  
• Buy Siegward's armor from Patches. This is done by either meeting Patches in the Cathedral and buying them there or by meeting him in the tower behind Firelink (need 20k souls for the key and more for the armor). After you get the armor, return to the well and give it to Siegward. You can kill Patches for the armor, but he sells some good items and gives you two gestures, so I recommend you leave him alive. You can forgive him, as long as you say "You know who I am" and will still get the gestures. If you meet him in Firelink, DO NOT forgive him, because then you will not get the prostration gesture. (credit to @Lucatiel for pointing this out, more info in the comments)
  
• Longfinger Kirk will invade in the Cathedral. After he is beaten you can loot his body in Rosaria's bedchamber but you must kill him BEFORE killing the Deacons of the Deep. DO NOT directly involve yourself with Rosaria's Fingers just yet. You can join to get the covenant item and achievement, but do not offer her any Pale tongues aside from respecs or changing your appearance, I'll get to why in the next section, but tldr it will abort Sirris' quest line.
  
• Meet Anri and Horace in Firelink after clearing the Cathedral.

The Abyss Watchers are a hard boss fight, but nothing is harder than finally beating them and finding out half your quests are aborted, which can actually happen if you've been postponing some things until later.

• Get your 5 "free" level ups from Yoel now if you haven't already done so. The hollowing is really annoying, but it is necessary if you want to complete Yuria's quest line and for some gestures. It gets more expensive every time you level up, but DO NOT heal the dark sigils. You can do this by giving the fire keeper soul to the firelink firekeeper and paying for your next 5 level ups, but this will abort Yuria's quest line and lock you out of a possible Anri ending.
  
• Give Dream Chaser's Ashes (illusory wall near the Farron wolf bonfire) to the Firelink Handmaid, this cause Sirris to appear again in Firelink and she will now ally with you. You will also get her gesture, which you will need to join the darkmoon covenant.
  
• Before the Abyss Watcher's fight, summon Gotthard and the Londor Pale Shade to get their gestures. Pale Shade is only available if Yuria has appeared in Firelink and is aligned with you. Use the dried fingers to reset the summon cooldown to summon two phantoms (can be bought from the handmaiden).
  
• Give at least one sorcery scroll to Orbeck if you've recruited him. If you don't, he will leave after Abyss Watchers. The Golden and Sages scrolls are found in the swamp before Abyss Watchers, so make sure you find at least one and give it to him before fighting the boss if you want to finish his quest line. If you didn't recruit him, don't worry about this, he'll still be in the same place by the Crystal Sages until right before the final boss.

As mentioned in the previous sections, you cannot complete Sirris' quest line if you have joined Rosaria's fingers, so make sure you don't join the fingers, because you can still complete the fingers' quests AFTER Sirris' is complete, so if you want to do as many quests as you can, you have to do Sirris' first. There is a small tradeoff for this though. You can't finish Sirris' questline until after Abyss Watchers, but there is a Heysel summoning location in the Farron Swamp before the legion doors to get to the Abyss Watchers which will give you a gesture like pale shade and gotthard (if you killed in the crucifixion woods invasion location). For it to appear though, you need to offer a pale tongue to Rosaria. I think you can see where this is going; because she is a summonable npc, you won't be able to summon her after beating Abyss Watchers, which you have to beat to finish Sirris' quest line, which you would need to finish before offering a pale tongue to Rosaria. Even though Sirris has sworn herself to you now and gave you an emote, if you offer a pale tongue to Rosaria she will turn hostile and ends her quest. I have not personally tested this, but it is pretty well known as occurring. If you want to test this, make sure to save your game file first in case you want to go back. If you find anything, please leave a comment to help others using this guide.

After the first few bosses, the amount of quests and dark spirits that you will miss due by killing bosses sharply declines, so I've combined them into one final section.

Wolnir
• In the Catacombs, meet Anri again. Also, while you're there, Tsorig can invade as a dark spirit. Beat him for an item and do this BEFORE killing him for real in Smouldering Lake AND before beating High Lord Wolnir. You have to do this before Wolnir, but I'm not sure if it has to be done before you kill him as an npc, I say do it before just in case. If you have trouble, open the gate where he invades with the lever and break the bridge while he's on it. If he survives, climb partly down the ladder and plunge attack him to finish off what's left of his health.
  
• In Smouldering Lake kill both Horace and Tsorig. Find Anri again in the Catacombs and tell about Smouldering Lake for the ring of the evil eye.
  
  
• If you have recruited Cornyx, you can summon Cuculus to help you fight the Old Demon King. Beating it with Cuculus still alive will allow you to get Cornyx's set from where you recruited him in the Undead Settlement. Take note that the Old Demon King is entirely optional, and there is no real time limit so you can do this as late as before heading into NG+. It's not very hard to do this if you summon Tsorig for this fight, as he will tank most of the damage until phase 2, at which point the demon king starts charging his moves up giving you plenty of time to attack him during. You have to beat Tsorig in his invasion spot before Wolnir if you want to do this.
  
• Reach Irithyll Boreal Valley's second bonfire, after which return to the bridge entrance and find Sirris' call to summons sign and help her beat Creighton. Speak to her later in Firelink for the silvercat ring.
  
  
  Pontiff Sulyvahn and Irithyll Dungeon
  
  
• Meet Anri again in Yorshka's Chapel in Irithyll for a gesture. DO NOT hit any of the statues, as this will mess up the ending for Yuria's quest line. After which explore the graveyard outside where Creighton can invade. Beat him and afterward, return to Irithyll's bridge entrance again to find his body and armor set on it. This must be done BEFORE beating Pontiff Sulyvahn. This can still work out sometimes if you've been following Patches' "questline" as he will save Greirat. To do this, after being tricked DO NOT kill him and talk to him at Firelink when he gets back before you kill the Pontiff.
  
• BEFORE beating the Pontiff and BEFORE heading to Irithyll Dungeon, send Greirat to plunder Irythyll and then meet Siegward in the Irithyll manor kitchen and share another toast. After which beat the Pontiff and Geirat will have returned, THEN you can head to the dungeon. Entering early will cause Greirat to die, so make sure you don't do this. If you already entered the dungeon, you can still have Patches save him. According to @Lizzardtong, you have to talk to Patches and tell him where Greirat went (Patches will disappear after reloading). Defeat a boss of your choice, and both reappear in the shrine. The method he mentioned requires that Patches still has Siegward's armor set. The dialogue option to talk to Patches about Greirat after sending him out is not available if you've bought ANY of Siegward's set. You have to let Patches live and get back to Firelink to do this. If you use this method, you can enter Irithyll dungeon without risking Greirat's life as long as Siegward is stuck in the well. Make sure you come back to him before Yhorm, as you MUST free Siegward before Yhorm. Just don't forget to speak to Patches before beating a boss after sending Greirat out or he will die.
  
  
• After beating the Pontiff and opening the way to Anor Londo, speak to Yuria and she'll spout off some stuff about getting married to Anri. Go to Darkmoon Tomb to conduct the ceremony, after which you can get the Chameleon spell from the Pilgrim presiding over it. Return later to get Anri's sword (weapon that scales with luck). You can speak to her before beating the Pontiff, but you will not be able to start the marriage.
  
  
  After Aldrich and Before Yhorm
• After beating Aldrich, a unique Green Blossom should appear in the Shrine Handmaid's stock. Return to the Undead Settlement and find another call to summons from Sirris outside of the Curse-Rotten Greatwood's boss room. Help her out again and then meet her at Firelink and have her swear loyalty to you. You don't have to do this before Yhorm, but you do have to do this BEFORE clearing the Grand Archives. Once this is done, you can find Hodrick's shield in front of a tombstone outside Firelink and his armor set in the Pit of Hollows.
  
• Go through Irithyll Dungeon and reach the Profaned Capitol. Get all the keys you can and free Karla from the Dungeon and do the same for Siegward in the Capitol. Make sure to free Siegward BEFORE fighting Yhorm.
  
  
• Once the way to Lothric Castle is unlocked, you can send Greirat to plunder it but as far as I know he WILL die this time and his body will be found in the Grand Archives. Give his ashes to the handmaiden and you'll be able to buy all his old ware plus some new goodies.
  
• Once the Grand Archives are cleared you can find Sirris' body and talisman at the same place you found Hodrick's shield. NOW you can finally offer Pale tongues to Rosaria's Fingers and progress the quest line.
  
  
  Final Boss
• BEFORE beating the final boss, make sure to buy all the stuff you want from Yuria. Also make sure to have recruited Orbeck, and then talk to Yuria about assassinating him (you actually don't need to do this). As you buy more spells from him he will give you rings and a gesture when you talk to him. After getting the Slumbering Dragoncrest Ring, buy all the spells and he will leave. You can later find his body at the Grand Archives; deliver his ashes to Yuria to get the Morion Blade (Weapon with Red Tearstone effect).
  
• Before the final boss you can summon Yuria (and Londor Pale Shade) to help you. If Yuria survives the fight, you'll find her items where she would normally be in Firelink. You can actually leave the arena and get them before triggering the ending.

These are things that will not break if you kill bosses early:

• You don't need to kill Old Demon King at all (optional boss), but I like to do it when I get to Smoldering Lake and summoning Cuculus and Tsorig before Wolnir. Tsorig can die, but keeping Cuculus alive will give you you Cornyx's set (check section on Wolnir for more info).
  
• After joining Rosaria's Fingers give her a Pale Tongue and then talk to Leonhard, who will appear in the bedchamber's entry door. Leave then return and find the Black Eye Orb. Go to Gwynevere's old bedchamber in Anor Londo and invade Leonhard there. There is no time/boss constraint for starting or finishing this questline (it does not break after Greatwood or Wolnir confirmed by @ebu leheb gaming in the comments) I'm pretty sure you can finish the entire game and still complete it afterward, but I have not personally tested this since I invade for fun and embers at low levels so I try to get the uncracked orb as fast as possible (I suck so I burn through the cracked orbs).
  
• Walk the "Path of the Dragon" at Archdragon Peak. Make sure to avoid triggering the Bell that calls the Nameless King until after you gained all the items from the dragon altars. After which, talk to Andre in Firelink about Hawkwood. Go to the Abyss Watchers' boss room to fight him for one more dragon item.
  
• Kill the Havel Soldier in Archdragon Peak. Afterward, go to the high bridge in Farron Keep and find the Havel set on a body near where the demon was.
  
• Give Irina all non-Londor/Deep braille tomes and buy everything from her. She will move to the bottom of the tower outside of Firelink. You can also find Eygon's body in her old cell in the Undead Settlement and get his weapon.
  
• You can give Karla the Londor and Deep braille tomes, just keep trying until she takes them. DO NOT give these to Irina, because buying dark miracles from her will cause Eygon to take her away. If you accidentally give her a dark tome, you can still avoid negative effects as long as you don't BUY any of the miracles. You will get all the tomes you accidentally gave her at the end of her questline.

The main things you miss out on with this route are:

• Anri's quest endgame, but you miss out on no items this way.
  
• Irina's "bad" outcome, again there is nothing lost from doing things right.
  
• Summoning Yellowfinger Heysel in Farron Keep for her gesture, which only works if you ally with Rosaria's Fingers. (can't get Gesture Maestro on first playthrough)
  
• The Londor Pale Shade invasions and items from them. If you make Yuria hostile or kill her, the Pale Shade will invade in both Farron Keep and Irithyll Boreal Valley.

Yellowfinger Heysel's emote is the biggest loss here, but you just have to kill Gundyr, Vordt, her in crucifixion woods, and Sages. After that you can fight her red spirit in Farron to get her emote in ng+. Assuming you didn't miss any emotes in new game, this will get you the the Master of Expression emote. Make sure to double check you didn't miss any of the late game emotes before going into ng+. The early ones you can always get in ng+, but if you're just going for the emote you won't want to fight the bosses all over again.

Dark Souls as a series is known for having very convoluted quests, and I don't say that lightly. Finishing them is quite the achievement, so pat yourself on the back for doing a great job!

Once again, a huge thanks to fextralife user Dapiek_Absaroka for the information in this guide. A lot of the information comes from him and this guide would not be possible without him. I just reformatted it and added supplementary information from my personal playthrough here and there.
thread: https://fextralife.com/forums/t56926/how-many-side-quests-can-you-complete-in-1-playthrough/

Thanks to @ebu leheb gaming for confirming that you can still start the Rosaria questline after Greatwood and Wolnir

Thanks for @Lucatiel for pointing out that you don't have to talk to Siegward to trigger the Patches encounter in the Cathedral of the Deep. Also according to his experience, if you meet him in the Cathedral you can say, "you know who I am" to him and receive the prostration gesture. Forgiving him (IN CATHEDRAL) doesn't have an outcome on receiving the gesture (but Dark Souls is an unforgiving world so screw that). IN FIRELINK you can't forgive him, because then you will not get the gesture.
I believe what breaks Siegward is killing the Deacons, but I may be wrong about this. Also, you can kill Patches to progress Siegward's quest, but you will miss both his gestures (and the stuff he sells is super good in PvP).

Thanks for @Lizzardtong adding more information about the Greirat/Patches interaction (the other way to save him in Irithyll). This is the only way to keep Greirat alive if you enter Irithyll dungeon before killing your next boss. Entering the dungeon triggers Siegward to walk through the dungeon, making him unable to save Greirat in Irithyll. However, if you have given Siegward his armor back (bought it back), Patches will be UNABLE to save Greirat as he needs to disguise himself to save Greirat (guess he's shy or something).

Also thanks to all the other users in the comments for being polite and helpful!