All cars:
Press Triangle, Square(2), X, R1 at the options screen. A sound will confirm correct code entry.

Truck:
Select any car, then hold L1 until the race begins.

Buggy:
Select any car, then hold R1 until the race begins.

Hidden car:
Select any car, then hold R1 + R2 until the race begins.

Various cars:
Select any car, then press Down at the transmission selection screen to cycle between a low rider, bus, rocket engine car, taxi, and police car. Note: The number of cars that are selectable will vary depending on how many keys have been found. To drive the police car, circuit mode must be completed.

Additionally, press Circle at the car selection screen to access four more vehicles, including a Viper, hippy van, VW Beetle, and McClaren.

UFO:
Hold R1 + R2 + L1 + L2 at the car selection screen and choose a car. Keep those buttons held and press X at the transmission selection screen. Then, hold Triangle until the race begins.

Alternate car colors:
Press Square at the car selection screen.

Reversed tracks:
Complete circuit mode to unlock an option for forward and reversed tracks.

Stunt mode:
Complete tournament mode to access a stunt mode option on the Golden Gate track.