Beat Story mode consecutive times, and get hidden characters. To access, highlight Squall and press the correct button:
3rd time- Cid (in the tank from Chocobo's Mysterious Dungeon 2), press L1
4th time- Mumba (secret Guardian Force from FF8), press L2
5th time- Cloud (main character in FF7), press R1
6th time- Cactuar (a.k.a. Cactrot, from many FF games), press R2
7th time- Aya (from Parasite Eve), press L1+L2
8th time- Original Chocobo (non super-deformed cutesy style), press R1+R2
9th time- Highwind ff6 (non super-deformed cutesy style), press L2+R2
10th time- Running man (dunno), press L1+R1
Bahamut class
Finish in first place in every stage (including the Final Fantasy 8 track) in the Chocobo and Behemoth class in grand prix mode to unlock the Bahumut class.
Mirrored tracks
Finish in first place in every stage (including the Final Fantasy 8 track) in the Bahamut class in grand prix mode to unlock the mirror mode selection on the options screen.
Quick start
Immediately before a race begins, hold Square + Up.