There are a total of 13 cars for you to acquire throughout the game, 4 of which you start with. The video below shows all 13 cars in the following order:
1) Jacky's car: Start in Civitavecchia and head to Siena. Choose "Shut up. Let's take a look around, Gorilla", and then "Well unfortunately, ours just broke down...". Then, win the race.
2) Gorilla's car: Start in Civitavecchia and avoid the police or win the race driving Angelo's car.
3) Starter 14) Valerio's slow car: Start in Bracciano and win the race against Valerio in an older/slower car and choose to take his car.
5) Starter 26) Valerio's fast car: Start in Bracciano and win the race against Valerio in a sporty car, then choose to take his car.
7) Starter 38) Renault 4: Start in Bracciano and lose the race. Choose to drive with the priest in Siena.
9) Starter 4 (using this causes Valerio to drive his bigger/slower car)10) VW Bus: Start in Bracciano, then turn left towards Piombino.
11) Innocenti Mini: Start in Civitavecchia and rob the bank with Angelo, then head straight to Piombino to steal a new car. Make sure to escape the police too (don't let them get ahead of you).
12) Trike: Start in Bracciano and lose the race against Valerio. Keep Olga with you and take the farmer's trike.
13) Police Car: Unlock 10 different endings.
Video courtesy of KugareNOTE: If you beat Valerio, be sure to select the option to keep his car in the intermission. Thanks GREGO McGEE!