Buddies basically function the same as in the previous game, with the exception that now buddies have vignettes which are playable asides. As you bring your buddy to different places they may try to get your attention- this is either going to give them a stat increase or a character vignette. Doing all the character vignettes and completing your buddy's questline will help you unlock the "X saved the world" achievements in the final chapter. You can achieve multiple of these achievements in one playthrough, but it will require more work on your end.

PS: if you have chosen to beat the game using your shadow taint (getting super-charged), you are locked out of getting the "saved the world achievements" for that playthrough.

Gabby: Everyone starts with gabby. Combo physical damage dealer and healer.
To get her ending, bring her with you around Crystaldream Lake, and she'll eventually want to wander off to the mushroom cave, then return. Keep exploring the area, and by the next day she'll return to the cave to check on her clones. Talk to all the clones, then return. Finally, get Gabby's last skill by wandering around Gray County, and you should get the ending. Gabby also sometimes gives you taffy and meat when travelling.

Obie: You find him in Goldwraith park and if you convince him to leave (not using a skill), you can find him again at the hobo camp where he'll come with you. Deals hot damage and has a physical attack that causes bleeding.
First take Obie to The Big Moist to get his first vignette, then back at S.I.T you can unlock his second one along with his final skill- HOWEVER you must also beat the flute minigame during this section, and cannot simply leave after finishing your business to unlock his ending (thank you Pyro Clown). Obie will sometimes busk to gain a few meat on your travels.

PS the flute minigame sucks and is annoying, but here are some tips"
-the demon man will repeat the section he played if you wait a bit in case you miss it
-the notes coming out of his flute go in the direction of the arrow key you want to hit
-you can play the notes as he's playing them if you have bad memory (this is better done on mute)
-GOOD LUCK

Molly: If you choose to accept the mob's offer, you'll find her on the street outside the antique shop. Say yes when she asks to come with you, and voila. Fires off several random rounds and also has a physical attack which decreases the target's muscle.
For Molly, bring her with you to Gray County where she'll visit her old home to get her final skill during her first vignette, where you should also pick up a letter found on the ground to continue her storyline. Later on, back by Ocean City, she'll ask to go pick up a cake and needs some money to buy it. Let her, and when you arrive in the government down, remember to visit the jail to drop it off for Nancy. This will unlock Molly's ending.

Barker: Take the botany course at S.I.T, and when you arrive at Prof. Kalos' vacation home, find the dog dish in the kitchen and fill it with water. You can then offer it to Barker, and he'll come with you. Has a standard bite attack and can lower mysticality.
Barker kind of just gains stats randomly and will sometimes throw up weird plant goo (which you can harvest as a cryptobotanist). Take him back to Crystaldream lake at somepoint for his first vignette. Barker has one other vignette when you arrive at Gray County which also unlocks his third skill (dealing poison to all enemies), and if you've been taking care of him up to this point you should be good to go.
For Barker's ending, you have to talk to the president inside the rift first then close the portal for good.

Simone: Take the robotics course at S.I.T. Once you pass, you can offer for her to come with you and she'll accept. Will deal damage to two random enemies, or set an enemy on fire for 3. Simone also comes with a little robot buddy which does a basic attack each turn.

Originally posted by Ccstewy9000:

When you reach Gray County, Simone will have her first Vignette immediately upon entering the area. She explores an oil rig set up, criticizes it a bit, and acquires an upgrade that doubles the amount of On Fire that her soldiering iron does.
Then shortly afterwards, she has another vignette that involves going back to her old house, being Chekov's house from the prologue, now mostly emptied of stuff. After a long puzzle requiring Simone's tools, talking to clarence for a combination, grinding down her tools, and repeated inspections of the mounted gun, Simone will be able to take it off the wall and acquire her last skill, which is, of course, named.... Chekov's Gun.
The skill itself just does single-target physical damage that scales off her Moxie.

To unlock the last vignette you have to go around Gray County and then back to Ocean City.

Adams: Take the chemistry course at S.I.T. You'll encounter him by pouring bronze revivifying solution on the Adams statue as you take the course, and he'll join your party, helping you complete the course. Makes potions that can heal or inflict poison.
Adams first needs to retrieve his binder from S.I.T which unlocks his third skill, and then bring him to Crystaldream lake for his other vignette. You will also need to make a vial of Umbraline Anhilating fluid (1-2-3-4) in the chemicals lab to use on the final boss. Yes, the solution to saving the universe is 1-2-3-4, beautiful isn't it? He'll sometimes make potion items for you as well.

Alphonse: Wandering around the Big Moist while carrying the Rift Crystal from Zimmer's house should have you encounter Alphonse, and he'll join you. Can decrease all the stats of a target, do a vampiric bite attack, or deal spooky damage to all enemies.
For Alphonse, he'll have two vignettes around The Big Moist where he'll ask to get back his shades from another gatorman, and another where he runs off to visit his family. Finish both then return back, and bring him with you to fight the president, where you can obtain his ending.