Successfully complete one of the following tasks to get a trophy:

88 Miles per Hour (Bronze): Travel through time.
All Stages Clear! (Silver): Finish the game.
Dr. Eggman Got Served (Bronze): Destroy Dr. Eggman's final machine.
Heavy Metal (Bronze): Defeat Metal Sonic without getting hurt.
Just in Time! (Silver): Complete the Time Attack mode in under 25 minutes.
Just one hug is enough (Bronze): Get a hug from Amy.
King of the Rings (Silver): Collect 200 Rings.
Paradise Found (Bronze): Complete a zone in the Good Future.
Saviour of the Planet (Silver): Destroy all the robot teleporters and Metal Sonic holograms in the past.
Statue Saviour (Bronze): Find the angel statue in Wacky Workbench.
Take the High Road (Bronze): Pass the upper Signpost in Collision Chaos Zone 2.
Treasure Hunter (Gold): Collect all the Time Stones.