Play ten multi-player races using a single cartridge to unlock the 00-Agent car. Keep playing in multi-player mode to continue unlocking bonus cars, in the following order: Red Shirt Rage, Pooka, Caddy Car, Galaga '88, Shy Guy, and Mario Racing. Win all the Platinum GP races to unlock all bonus cars in the game.
To unlock a car in Car Attack mode, win the indicated Grand Prix to get that car as an opponent in Car Attack mode. After defeating that car in Car Attack mode, you can then drive it in Car Attack mode.