To start all of the sidequests, its a pretty standard routine. meet any NPC during a run, (NPC's in the breach do not have sidequests) and talk to them whenever you meet them. as long as you keep chatting to them consistently during runs will advance their dialogue, however, the old man will need to be talked to to advance dialogue AFTER you obtained his chest. In case of Goopton, dont worry about not being able to read what he says, it should be obvious when he asks for his item.

NPC's with quests include:
-Bello (The Shopkeeper)
-The Old Man (the one that gives you a free chest after saying he wants to be alone)
-Old Red (the blue bullet)
-Flynt (the living lock, with his magnificent door.)
-Goopton (the one that speaks in moon runes)
-Daisuke (the NPC with the combo meter)

This means that the Nurses, Trorc, and both of the Minigame NPC's do not have quests.

Once you have talked to them enough times, all of the aforementioned characters should ask you to obtain a specific item.

Now, this is where the challenge starts.

All of the characters want an item.
Bello wants secure stools,
The Old man wants a gameboy,
Old Red wants his old gun Daisy,
Flynt wants Glue,
Goopton wants... whatever Goopton wants.
and Daisuke wants Nut butter and a Pretzel.

All of these items can be found in one place...
MINIGAMES.
Both minigames, (Just dessert golf and Winchesters original game) have balloons flying around with the Resourceful Rat's face on them. These balloons are carrying the required items. If you fire your golfball or your cannon at them and obtain the item you have it for the rest of the run.

DO NOT DESTROY THE BALLOON, doing this will cause the item to disappear and you wont be able to get it again that run, you have to get the item itself.

Now, its kinda sad concidering its based on RNG whether or not you will get a minigame room, but from what I have seen, the minigames will always have a rat balloon, as long as there is an active quest.

After obtaining the item, talk to the character giving the quest to give them the item, and the achievement should pop. (I didnt know where to mention it but Daisuke needs two items, which means his quest should be done over across at least two runs)

Lastly, you might be wondering: "wait, does this mean I have to get lucky to get both a minigame room AND the specific NPC character?" Well, no, Bello is obvious because he appears on every floor, so you should never miss out on him to give him his stools. And the other characters are guaranteed to appear if you have their quest item, so luckily the quests arent *THAT* RNG heavy.
The only way you are able to miss out on a quest during a run, is if the minigame room is on the last floor, because then there wont be any more floors for NPC's to spawn on.

This was the first guide I have made, so id be glad to hear any feedback you have.

Other than that, I hope this guide has helped in some way, shape or form, and cleared up any confusion about the character quests.

(If anyone is able to let me know whether or not the balloon appears in the new minigame added in the hello to arms update, that would be much appreciated!)

Have fun Achievement hunting!