Unlock Extra Characters
Ed - Beat the fourth level of Blizzard Castle.
Chick - Beat the fourth level of Dino Mountain.
Rolly - Beat the fourth level of Pirate Island

Unlock GBA Minigames
Hatch the bonus eggs in the following levels:
Dino Mountain level 2 - Billy Hatcher Hyper Shoot Easy
Sand Ruin level level 2 - Billy Hatcher Hyper Shoot Hard
Pirates Island level 2 - Chu Chu Rocket
Circus Park level 2 - Nights
Blizzard Castle level 3 - Puyo Pop

Hint: Find Sonic Eggs
Beat one world with every character, and then there will be a Sonic Egg in each level of that world

Hint: Getting Lives
Find an egg with a 1-Up item in it, and then keep hatching it over and over again to get a ton of lives.