You need two controllers for this code.
1. Turn on the game and press START on the contoller 1 until you see the screen that says "Push Start" (if you see "Grand Prix Mode", you're too far.)
2. On controller 2, press up on the D-pad, C-down, R, L, and Z all at the same time. Then release them.
3. Immediately press start or the A-button on controller 1 to enter the Grand Prix mode.
All tracks and cars should be available now.
Note: If you can't get this trick to work, do step 2 a couple of times. (Press all the buttons a few times.)
Credit Goes To: IGN 64