That's an interesting discussion I've had before. Waves in general only ever travel at one speed in a medium, which is defined by wavelength times frequency.
In a more practical real-life situation, the same thing happens with sound waves.
If something exceeds the sound barrier, it cannot be heard in front of it - two examples - bullets, or
here - planes.
Notice you only hear the plane as it goes past (more specifically, after it's sonic shockwave has passed). With lights, if you assume they shine 360 degrees (headlights don't, I know), then you would only see the lights after the car had past.
As they don't, I suggest not driving at the speed of light at night, because your lights won't illuminate the road, nor will pedestrians see you coming.
Also worth note is that frequency changes. Like a car horn is pitched up if it's approaching you, and down if it's going away, then car lights will become more blue the faster the car goes.
So for your answer, you'd have blue headlights that wouldn't be able to light up anything.
