All Cars:
At the cheat menu press ,X ,Triangle, , L2, .

All Tracks:
At the cheat menu press R1, , , , R2, L1.

Unlimited Turbo:
At the cheat menu press , , , , L2, L2.

More Drivers And Cars:
Press R1 at the driver select screen to get more drivers or L1 to get more cars

Fog:
At the cheat menu press R2, R1, , , , .

Big Head Mode:
At the cheat menu press , , , L2, L1, R2.

Thick Fog And New York Course:
Beat the game in demolition mode under the easy, medium, and hard difficulty setting to unlock the Thick Fog And New York track options at the cheat menu.

Rocket Wheels Boston Big Dig Course:
Beat the game in Block Your Opponent mode under the easy, medium, and hard difficulty settings to open up the Rocket Wheels and Boston Big Dig track options at the cheat menu.

No Time Out and Moon Course:
Beat the game in time trial mode under the easy, medium, and hard difficulty settings to open the No Time Out and Moon track options at the cheat menu.

Night Drive and Alpine Raceway Course:
Beat the game in lock-on mode under the easy, medium, and hard difficulty settings to open up the Night Drive and Alpine Raceway track options at the cheat menu.

Death Car And Miami Course:
Beat the game in simulation mode to open up the Death car and Miami track options at the cheat menu.

Infinite Continues Option:
Beat the game in season mode to open up the Infinite continues option at the cheat menu.

Jump Option:
Beat the game in last man standing mode to open up the Jump option at the cheat menu.

Driving 101 FMV Sequence:
Beat all Driving 101 options to open up the Driving 101 FMV sequence.

Thanks to Revolution readers Gromlok and Peter!