After you reach Level 3, you will start to get invites to car challenges.
Car challenges are special races where you get to test drive a new car, and everyone else drives that same car. When you get a new car challenge, it appears in one of the posters on the main screen. The 1st place winner of the race wins the car. If you don't get 1st place, you can retry the event for gems.
Upcoming car challenges can be seen on the Profile popup. Click the button with your name on it in the upper left corner of the screen to see an enlarged version of your level progress, and look for the icons with the key.
There are 2 other ways to get cars.
On the Team screen, you can browse locked cars and unlock them for a single gem purchase.
You can also win cars in the Daily Challenge. When you beat the Daily Challenge, you will see a reward wheel. If the wheel has a yellow ribbon on it, that's the grand prize and if you hit it you will win that car. Cars do not always appear on the reward wheel, however, so check back every day.