Okay, at the speed of light I'm not sure what would happen, maybe you'd infinitely increase in mass, or your time to another observer would completely stop... point is, these relativistic laws probably break down...
For the sake of this question, we will say close to the speed of light:
You will not notice anything strange happening in your own frame (everything that's going an insignificant speed relative to you, like your magical car, your trousers and shirt etc). Your headlights will work fine. However, you will notice that your surroundings that are whizzing past you at that speed will appear shorter in the same dimension as you are travelling. So if you travel north at something like 0.90*C, objects will have decreased in length in north direction by a factor of 0.43, so they appear 56% shorter.
What if someone else outside sees you? They'd see your car shorter by the same factor in the direction of travel the same way you see them, but everyone appears normal to themselves.
This does not happen with light, if someone outside your vehicle shines a torch in north direction, you wouldn't see that light trying to catch up with you, they'd see it at C (light speed). If they witnessed you put on your headlights, they would also measure the light speed to be C. This holds as one of the foundations of relativity, that the speed of light is the same to every observer in different reference frames.
Look up time dilation too.