The name of the game is simple: Collect Rings! In each stage, collecting rings will gradually restore the proper color and sound to the vicinity. Collect as many as you can!
Scattered around the Eggman’s base (the hub area) are a selection of 6 acts to choose from, accessed via Giant Ring portals (once in one, press A to enter the stage, and B to cancel)
To get to Eggman’s Machine, Sonic must collect the 6 chaos emeralds; the Giant Ring portals are located in the upper area of the hub, and can only be accessed by clearing acts with at least 64 rings (An “S Zone” token will appear at the start of the hub is done successfully). After obtaining all 6 emeralds, the teleporter on the far right side of the HUB will be accessible.