This is how to unlock different suits for characters:

Suit	How to unlock
Espio's Knight Suit	Win Espio Cup in Single Player
Espio's Mummy Suit	Beat silver 20 times with Espio
Espio's Ninja Suit	Win Espio Cup in Single Player
Espio's Race Suit	use Espio's Signature Move 20 times
Knuckles's Armor Suit	Win Knuckles Cup in Single Player
Knuckles's Eggman Suit	Win Knuckles Cup in Single Player
Knuckles's Gladiator Suit	Beat Rouge 20 times in any mode with Knuckles
Knuckles's Race Suit	Attack any rival 50 times with Knuckles
Metal Sonic's Circuits Suit	Win Metal Cup in Single Player
Metal Sonic's Jester Suit	Beat Tails 20 times with Metal Sonic
Metal Sonic's Mach 3.0 Suit	Successfully copy every Rival's Signature Moves with Metal Sonic
Metal Sonic's Mach2.0 Suit	Win Metal Cup in Single Player
Rouge's Armor Suit	Beat Knuckles 20 times with Rouge
Rouge's Race Suit	Collect 500 Rings with Rouge
Rouge's Raider Suit	Win Rouge Cup in Single Player
Rouge's Stealth Suit	Win Rouge Cup in Single Player
Shadow's 80's Suit	Win Shadow Cup in Single Player
Shadow's Flame Suit	Win Shadow Cup in Single Player
Shadow's Mercenary Suit	Beat Sonic 20 times in any mode while using Shadow
Shadow's Race Suit	Use Shadow's Chaos Control 20 times
Silver's Christmas Suit	Win Silver Cup in Single Player
Silver's Cyber Suit	Beat Espio 20 times in any mode with Silver
Silver's Lightning Suit	Win Silver Cup in Single Player
Silver's Race Suit	Use Silver's Signature Move on any Rival 20 times
Sonic's Black Tie Suit	Beat Shadow 20 times in any mode with Sonic
Sonic's Cheetah Suit	Win Sonic Cup in Single Player
Sonic's Ice Suit	Win Sonic Cup in Single Player
Sonic's Race Suit	Run 15,000 meters with Sonic
Tails's Ace pilot Suit	Log 3 minutes of fly time with Tails
Tails's Captain Suit	Win Tails Cup in Single Player
Tails's Tiger Suit	Beat metal sonic 20 times with Tails
Tails's Zebra Suit	Win Tails Cup in Single Player