A: If your car could travel at the speed of light, would your headlights work?
A: Yes, er, no, er, wait a minute! This is a trick question. I don't like trick questions. I think you must be one of them dad-burn pinko-commie-hippie-trouble makers. Get offa my lawn!
Actually, yes they would. There is no absolute velocity. There is only relative velocity. The light given off by things coming at you is blue shifted, light from things going away is red shifted. When light is blue shifted the apparent wave length goes down and the frequency goes up. Just the opposite happens when it is red shifted: the wave length gets longer and the frequency gets lower.
On earth, shoot, even in the solar system, relative velocities are too small to make a noticeable difference. But you get going really fast and it does start to become noticeable. When they talk about the red-shift of light from stars they are talking about measuring how fast the stars are going away. If they were going really fast, the light would get shifted down into the infra-red or even the microwave region.
Things approaching each other will see the light of the other blue-shifted. If they were going fast enough the light would shift into the ultra-violet range, then X-rays and finally gamma rays.
So if you were driving down the road at the speed of light, everything in front of you would be being blasted by really powerful gamma rays. Any light reflected back at you would be the same. Put on your super gamma ray goggles. Any light from your taillights would be so red shifted as to be undetectable.