To unlock other cars, you need to find three keys per level. When you start a level, it is a good idea to find all the keys, then complete the laps. It does not matter what you place when you are looking for keys; take your time, and complete the level. If you do not complete the level and located all the keys, do not worry; you still get the prize. However to go onto the next level, you must place third or higher.
During the third Grand Prix, in the second to last level "Tomb And Gloom" is where you can unlock the find fastest car once you have found all the keys. The third and final key is hidden well here. It is in a curtain draped off the ground in a descending matter (just hop of the ledge onto the curtain). A good rule of thumb when looking for keys is to change the cars view to a more elevated position, then try to spot the sparkling keys in the distance.