Sounds like fun, let's give it a go.
Sonic 1 --> Sonic and Eggman's first face-off. Both 16-bit and 8-bit games are different tellings of the same event.
Tails' Skypatrol --> Tails being Tails before meeting Sonic or Eggman.
Sonic 2 (8-bit) --> Eggman kidnaps Tails, Sonic rescues him. Eggman realises the only thing that would beat Sonic is a robotic version of himself, develops Silver Sonic as a prototype.
Sonic 2 (16-bit) --> Eggman fixes and improves Silver Sonic, also builds giant Eggman mech.
Sonic 3K --> Eggman hears rumours of powerful Master Emerald, tricks Knuckles into thinking Sonic is the bad guy. Upgrades Silver Sonic to Mecha Sonic, upgrades giant Eggman mech.
Sonic & Tails --> vs. Eggman again.
Sonic & Tails 2 --> Eggman tricks Knuckles into thinking Sonic is the bad guy again. Fang decides to be an opportunist. Knuckles finally accepts Sonic as the good guy (for a while).
Sonic 3D --> Tails and Knuckles helps Sonic find Flicky Island's Chaos Emeralds. Eggman upgrades giant Eggman mech again, continues working on new robot Sonic behind the scenes.
Tails Adventure --> The trio part ways. Tails spends more time working on his technological abilities to help Sonic defeat Eggman, finds his own rival instead.
Sonic CD --> Sonic saves Amy, spends rest of life trying to run away from her. Eggman unveils his new, perfect robot Sonic: Metal Sonic.
Sonic 4 --> Don't whine, it fits here. Eggman recovers Metal Sonic, Tails returns to Sonic to help defeat him. Eggman upgrades Metal Sonic and giant Eggman mech again.
Chaotix --> Eggman decides to use Metal Sonic against Sonic's friends instead. Upgrades Metal Sonic again. Sonic and Tails eventually give chase, but everything's sorted by the time they get there.
At least, that's what makes most sense to me.
EDIT: Swapped a couple I wasn't happy with.
EDIT2: Grammar fix.
This post has been edited by DigitalDuck: 30 November 2014 - 07:51 AM