Non-Spoiler List of Quests and How They Start

Quests: In Order of Availability

50 Ways to Die at Dr. Nick's: (Puzzle/Mystery Quest)
Start - AutoStarts when you load your game

Barking Bad:
Start - Talk to Dr. Nick at any time

Settling an Old Score:
Start - Talk to Mia after receiving "Another Clue" from the Vend-O-Matic in the Puzzle Quest

Good Screws, Everyone:
Start - Talk to Farnsworth after unlocking the Back Room at Dr. Nick's

The Projects - Part I:
Start - AutoStarts after completing Good Screws, Everyone. Will not progress unless or until you have completed Barking Bad

Physician, Heal Thyself:
Start - AutoStarts after completing The Projects - Part I

The Projects - Part II:
Start - AutoStarts after completing Physician, Heal Thyself

The Quizling:
Start - AutoStarts after completing The Projects - Part II

Pain & Fury:
Start - AutoStarts after completing The Quizling

Dr. Nick's - Epilogue:
Start - AutoStarts after completing Pain & Fury