Get cheat menu
Unlock any cheat then go to the options screen and you should see a cheat menu.

Get Custom Series Championship
Get gold medals in all races. Finish the entire Championship mode to get the Custom Series.

Get free-run mode
Finish the entire Custom Series championship. Free-run mode should now be available from the cheats menu in options.

Get invincible car mode
Finish the Grand Prix Championship with all gold medals. The cheat will now be available in the cheats menu under options.

Get runaway mode
Get gold medals in every challenge on your way to completing crash mode. Now go to the cheat menu for runaway mode.

Unlock New Cars
To unlock the cars below, complete the corresponding goals:

Gangster Car: Beat Pursuit 3.
Hot Rod Car: Beat Face Off 1.
Japanese Muscle Car: Beat Face Off 2.
Oval Racer: Beat the car in Face Off 2.
Polic Car: Beat Pursuit 1 and destroy the villain's car.
Super Car: Beat Face Off 4.
Custom Compact: Beat Custom Series Qualifier.
Custom Coupe: Get all gold medals at Split Second Grand Prix.
Custom Muscle Car: Beat Pursuit 6.
Custom Pickup Truck: Beat Pursuit 5.
Custom Roadster: Get all gold medals at the Point Of Impact Grand Prix.
Custom Sports Car: Get all gold medals at the Speed Streak Grand Prix.
Custom SUV: Beat Pursuit 4.
Classic 1970 Car: Destroy the car with a police car in Pursuit 2.
Drivers' Ed Car: Get all gold medals in Driving 101.