Actually, Einstein worried about similar problems and this is how he came up with the idea that nothing can travel faster than light. If the car is traveling as fast as light, then the light coming out of the car will not travel faster than the car --- in fact, the beam of light will look as though it has been pressed down to the car. In other words, the light will not come out.

Well, it's quite hard to have a car go at the speed of light, impossible according to the special theory of relativity. (I'm not going to get into the general theory of relativity because that's a more confusing issue.)

Here's the easy way to understand why: remember the famous equation E = m c ² ?
E is the energy of the car, m is the mass of the car, and c is the speed of light. Really this equation isn't quite explanatory enough.

Let's say the mass m is the mass of the car when it is at rest, or "rest mass" . Then the equation is really

E = m c ²/(1 - v²/c²)^(1/2)

(note that c and v are to the 2nd power, so the power(1/2) is the square root of the stuff in parentheses). Here v is the speed of the car. So you can see that when v gets close to c, the denominator gets close to zero, which means that the energy gets very, very big. In fact, the energy is infinite if you try to go at the speed of light.

On the other hand, if your car went very close to the speed of light past me, I would see the light coming out of your headlights at the speed of light. You, sitting in your car, would also see the light come out at the speed of light. This is what relativity tells us!

Actually, thinking about this question (a very similar one) is what caused Einstein to invent special relativity. You can't go the speed of light without expending infinite energy, but you can thing about going very close to that speed. (Say 1 MPH slower than C).

If you turn your head lights on, you will see light traveling at the speed of light anyway-- but don't try to look for your lights to shine on the signs you are going by -- every thing except for things directly in your path will appear to be behind you and shifted strongly to the red (in fact at the speed you are going, you can only see short gamma-rays coming from behind as light. On the other hand, in your car, your lights will work as usual-- even tossing a ball forward will look normal to you. From the outside, your headlights will appear as a burst of gamma-rays as you go by... and your ball will be traveling a tiny bit faster than your car, but still less than the speed of light...