Successfully complete the Babylon or Hero story. When that storyline is played again, you can choose any stage and play in Mission mode.

World Grand Prix mode

Successfully complete the Hero story to unlock World Grand Prix mode.

Bonus tracks

Successfully complete the indicated task to unlock the corresponding track:

80s Boulevard: Successfully complete the Babylon story.
90s Boulevard: Successfully complete the Babylon story.

Bonus characters

Successfully complete the indicated task to unlock the corresponding character:

Amigo: Successfully complete all missions in Story mode.
Billy: Successfully complete all missions in Story mode.
Blaze: Successfully complete the Babylon story.
Cream The Rabbit: Successfully complete the Hero story.
Eggman: Successfully complete the Babylon story.
NiGHTS: Successfully complete all missions in Story mode.
Rouge The Bat: Successfully complete the Hero story.
SCR-GP: Successfully complete the Babylon story.
SCR-HD: Defeat SCR-HD in a Free Race.
Shadow The Hedgehog: Successfully complete the Hero story.
Silver: Successfully complete the Babylon story.

Bonus gear

Successfully complete the indicated task to unlock the corresponding gear:

Angel/Devil gear: Successfully complete the Hero story.
Chaos Emerald gear: Earn an "Extreme" rank on all missions in Story mode.
MAG gear: Successfully complete all missions in Story mode.

Movies

Successfully complete the indicated task to unlock the corresponding movie:

Ending: Successfully complete the Babylon story.
Insert: Successfully complete the Babylon story.
Opening: Start the Heroes story.

Easy Rings

Enter Mission mode, and select Mission 2, "Collect 'X' Rings" on any of the tracks. During this mission, you are not limited to a set number of Rings that you can collect. Collect as many Rings as possible before reaching the goal, then wait for the results screen to appear. Once the results finish displaying, all the Rings you have collected will be automatically sent to your Ring account (although the game will not show you this). Pause the game, and select "Restart" to begin the mission again. Keep repeating the process. If you can get at least 250 Rings per try, you should have collected enough Rings to purchase even the Rainbow gear within an hour or two.