Waaaaaiiit . . . . . . something's happening on this board?
*Very minor spoilers if you haven't finished the first 10 or so races*
To answer your question: look at your in-game map, there will be a location where you can buy cars. There's one in each of the first three areas, I think you need to own 3 territories in your first area of the city before you can buy them. You'll need to drive there.