Earn enough notes during any stage to score a golden blossom and unlock its Fast Mode. Activate it by pressing Y, going to the stage you've earned Fast Mode for, then press X. Earning another golden blossom while playing Fast Mode will unlock the special image for that stage.
Locate all five Tori birds by "hitting" the eggs that appear in World 1-4, World 2-6, World 3-7, World 5-8 (unlockable after completing the game), and World 7-3 with the Y button. This will unlock the door in Symphony City and grant you access to the hidden world.