All cars:
At the main menu, press Up, Down, Right, Right, Left, Up, Up, Down

All tracks:
At the main menu, press Up, Up, Left, Down, Left, Right, Right, Up

Story mode:
At the main menu, press Down, Left, Up, Right, Right, Up, Down, Left

Arcade championships:
At the main menu, press Right, Left, Up, Right, Down, Down, Left, Left.

Multi-player championships:
At the main menu, press Left, Down, Left, Up, Right, Left, Down, Right

Thanks to Revolution readers Greg Colwell and levi!