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

Drive The UFO:
Go to the car select screen, hold L1 + L2 + R1 + R2. Keep holding these buttons at the transmission screen then press X. As soon as you're done with that, hold Triangle until the race and then you should have the UFO.

Secret Cars:
After selecting your vehicle -
Hold L1 until the race begins for a miniature truck.
Hold R1 until the race begins for the England Buggy.
Press DOWN on the Transmission screen for a taxi.