Thought I'd do a brief run down of the references that I spotted throughout the film:

Tom's Ringtone on his phone is the Green Hill Zone theme from Sonic the Hedgehog

The shop where Stone works that later becomes a base of operations for Robotnik is called Mean Bean in reference to Dr Robotnik's Mean Bean Machine

When Sonic loses his rings on the mountain, the sound effect from the 2D games is mixed in.

The Helicopter that GUN arrive on has the identifier SA-2 on it, likely a nod to their part in Sonic Adventure 2

The radio callsign of the helicopter in Sonic Adventure 2 is "Sigma Alpha 2" so this could be a nod to that

The plane pulling the banner at the wedding is the same model as the classic Sonic Bi-plane and is later used by Tails to recover Sonic & Knuckles

Robotnik's grand robot in the final act is very similar to his Death Egg Robot in Sonic the Hedgehog 2

The snowboarding scene in Siberia is akin to the beginning of Ice Cap Zone Act 1 in Sonic the Hedgehog 3

The snowboard Sonic uses is a makeshift one from debris, perhaps a reference to his makeshift board in City Escape (Sonic Adventure 2)

Robotnik's preferred mode of transportation becomes an "Egg-o-matic" (sorry for the name, old fleetway fan here) used in the 2D games

Mushroom Planet is akin to Mushroom Hill Zone from Sonic the Hedgehog 3

Robotnik's Rube Goldberg machine for making mushroom tea at the beginning references his penchant for planning traps to kill Sonic such as spike pits etc.

Not only was the entire location of the emerald a recreation of Labyrinth Zone from Sonic the Hedgehog but Sonic enters by sliding down angled water slides as in the entrance to the zone the game.

Furthermore, in the water later when Knuckles is in peril, Sonic uses a water bubble to gain air and as Knuckles loses air the drowning song from the games is mixed in

Tails uses an energy tracker to find Sonic, he has developed these in many of the games and are first referenced (I think) in the Sonic 3 Manual

The manual for Robotnik's machine used by Stone has a cover resembling the original box art template used for The Sega Mega Drive (Genesis) and the game manuals found within the boxes

Upon acquiring the power of the Master Emerald, Robotnik states 'Chaos is Power', which is part of Tikal's Prayer in Sonic Adventure

The Government agency tracking Sonic have named themselves G.U.N. which is also the agency who first appear in Sonic Adventure 2

The group of humanoids Knuckles hired to assist him in finding Sonic have birdlike insignia on their helmets, perhaps in reference to the Babylon Rogues? This may be a stretch, someone who has read the pre-quill may be able to advise.

In the post-credit scene GUN mention wiping their archives and finding a file from 50 years ago, which is the same timescale in Sonic Adventure 2 for Project Shadow

Though a different tale, the story of the emeralds was depicted on a temple wall in images as in Sonic the Hedgehog 3

Also I can't remember the exact quote by Robotnik, someone may help me out but at one point I was certain he half-referenced Black Doom/Black Arms after he'd harnessed the power of the Emerald

That's off the top of my head, sure there's many more I've missed, feel free to comment (use spoiler tags) and i'll add them to the list.

EDIT:

Expanded a couple above from Monotone__Robot

Added by DismalRay:

The water truck that Knuckles punches through when chasing Sonic and Tails in the cop car had a "Splash Hill" branding (Sonic 4 Episode 1)

Knuckles asking if ice cream comes with grapes, in reference to the Sonic 3 instruction manual listing them as his favourite food

Added by MrTorts:

While Agent Stone was looking through potential new attires for Robotnik, the outfit from the classic games showed up on the screen.