You need 2 controllers for this code...
Turn on the game and press Start on controller 1 until you see the screen that says "Push Start" (if you see "Grand Prix Mode", you're too far). On controller 2, press Up on the D-pad, C-Down, L, R, and Z all at the same time. Then release them. Immediately press Start or A on controller 1 to enter the Grand Prix mode. All tracks and cars should be available now.

When choosing car, press R. Each car has 2 colors, except the controller, which has 4 colors.

At the starting line, hold both the brake and the accelerator. When the starter says, "Go!", release the brake and your aerocar will rocket off the starting line.

Finish a race with the final 2 digits of your time being 64. This is really a matter of luck. Use this technique to get the other four hidden cars. NOTE: You must do it once for each car- you can't do it once and earn all the hidden cars.