After you have bought a car, on its upgrade screen you can see that all the upgrades are locked. By driving the car you will begin to make progress towards the first upgrade. This will be indicated by a progress bar, and at the end of each race where you complete another 10% of progress you will be notified on the post-race screen (for instance: "Upgrade alert: 10% progress - Advanced Setup").
Progress towards the upgrades is tied to miles driven in the car (similar to upgrades for your crew chief and engineers). I found the first upgrade to unlock fairly quickly - five rounds of a short Rallycross track in PvP was enough for my Ford Fiesta.