If you get 20 points in the Championship Mode, you can change your car's paint job on the car select screen by pressing L or R. Even better: If you get 150 points, you get two-tone paint.
You can power up your cars if you get a certain number of points in the Championship Mode, for example:
If you get 8, you can raise to power level 1 (press Up- or Down-C)
Get 100 points, and you get power level 3.