To become Super Sonic, simply find all Chaos Emeralds. Here's how to find them:

1. Emerald: Island course
(behind the door which opens with 50 rings)

2. Emerald: City course
(behind a door which opens with 20 rings, bottom of the course after falling down the hole in the street short after the tunnel)

3. Emerald: City course
(behind the door which opens with 50 rings)

4. Emerald: Ruins course
(behind the door which opens with 50 rings, location left side of the course on some arcades right after the start)

5. Emerald: Ruins course
(behind the door which opens with 20 rings, location is right in front of you after racing up the ramp)

6. Emerald: Factory course
(behind the door which opens with 50 rings, location is just after the looping the emerald will jump out of the machine in front of you)

7. Emerald: Factory course
(behind the door which opens with 50 rings, location short after the looping left of the ramp where a excelerator is located. The emerald will pop out of the submarine which waits for you right ahead)