Otherwise, the light from your headlights would be exceeding the speed of light as soon as you put the car in gear, and so would all the light striking from the front, relative to the surface it reflects off of. But that is not what happens, because the speed of light is constant everywhere regardless of scale, relative velocity of the particles involved, etc.
Exactly. The frequency of light changes, not it's velocity. That is why you have red-blue shift. There are astronomical bodied travelling at a significant percentage of the speed of light in relation to us, but the photons (light as particle) emitted by such bodies that reach us are still travelling at c, only the frequency of the light (as wave) is shifted to reflect the relative velocity of the source.We see this effect all the time in 2 way radio communications. It's called the "Doppler Effect" where a signal from a source travelling toward you is shifted slightly higher in frequency
[Whoops, looks like a few other people already brought this up.]