Light travels at the same speed, in all reference frames.
Therefore you will see light passing you at 300,000 km/s, while I too will see the light moving at 300,000 km/s. Neither of us would be wrong either.
you will see your headlights shining ahead of you, like usual, since to you, everything's normal. But the same is true for me who's standing still in relation to you. To me, light would be traveling at a constant speed of 300,000 km/s, which would lead to a thing called time dilation.
What this means, is that if I'm standing still, and you're travelling in a spaceship at the speed of light that is completely see through, with a mirror floor, we can do / experience the following.
If you were to stand with a torch, and shine it down to the floor, it would bounce straight back up. To me however, since you're moving forward at the speed of light, I would first see the beam of light when you switch on your torch, and then I would see it going down, while you're still moving forward (so it would look lik a V) as it bounces up and reaches you a couple of miles further. In othe words it's going down and forward, then up and forward as opposed to just up and down as you're experiencing it.
This means that to me, the light travelled much, much further in exactly the same timeframe. To you, in 1 second (just an example), the light travelled just a few feet, however according to what I saw, (due to the light moving in the shape of a V as you're moving forward), I would have seen the light travelling a few thousand miles.
We know that speed and distance leads to time (how long it would take you to cover the distance at your current speed), and since we know that the speed of light is the same to all observers, then time itself is what changes. So while you're seeing your light shoot off ahead of you, I will see you slow down, since your beam of light CAN NOT cover thousands (it can, but this is just an example) and thousands of miles in 1 seconds since that would mean it's travelling faster than it should. So you will slow down in my reference frame, and appear to be in slow motion. This is time dilation.
But in short, light will behave as it should on your space ship, and you will experience time as usual, since to you nothing changed at all. In your opinion, it's everything, and everyone else who's got it wrong...even though all of you are right.