Alternate Color
     Press Square at the Car Select screen.

Bonus Cars

Muscle Car
     Find 4 keys in any track.
 
Hot Rod
     Find 8 more keys in the other tracks.
 
City Bus
     Find all 16 keys.
 
Police Car
     Finish 1 player GP Mode.
 
Taxi Cab
     Finish 1 player Circuit Mode.

Secret Cars

UFO
     At  the Car Select screen, after choosing a track, hold L1 + L2 +
R1  + R2, then press X to choose a car. Continue to hold those buttons
on  the transmission screen and release after pressing X. Now, quickly
hold Triangle until the race begins.
 
Mini Truck
     Choose  your car, hold L1, then press X to select it. Continue to
hold L1 until the race begins.
 
British Mini
     Choose  your car, hold R1, then select your car. Continue to hold
it until the race begins.