Pause the game and enter the following cheats:

Activate cheat mode (do this one first!):
Up, Down, Left, Right, Down, Right, L1, R1, R1

Blood Mode:
L1, Down, R1, Left, L1, Right, R1

Angels (when blood mode is activated):
Up, R1, Down, L1, Up

Immortal:
Left, R1, Up, L1, Up, Right, R1, Down, L1

Choose Mission:
Up, Down, Circle, L1, R1, L1, Circle, Down, Up

Next Mission:
Square, Circle, R1, L1, Circle, Down, L1, Up

Car Select:
Left, R1, Right, R1, Left, R1, Right, R1

Debug Mode:
Up, Right, Left, Down, Circle, L1, R1, R1, L1, Circle, Down, Left, Right, Up

Invulnerable:
L1, L1, L1, L1, Left, Circle, Circle, Square, L1

Infinite Fuel:
L1, Circle, Left, L1, Circle, R1, L1, Up, R1, Down

Car Tune-up:
L1, R1, L1, Up, Down, Circle, Down, Right, Left, Square, R1

Extra Nitro:
L1, Circle, Down, L1, Up, Square, Circle, R1

Add 1 minute to time:
Down, L1, L1, Circle, Circle, R1, Up, Square, L1

Extra Money:
L1, R1, Up, Circle, Down, Square, Right, R1, L1

Access All Time Trials:
R1, L1, Circle, Left, Circle, Circle, Left, L1, Circle.

Access NY Time Trial:
L1, Right, Down, Left, Up, R1.

Access Tokyo Time Trial:
L1, Left, Right, R1, Left, Right, L1.

Access Subway Time Trial:
L1, Left, L1, R1, Right, R1.