The misunderstanding all comes down to this... If you are travelling AT the speed of light and turned on your "headlights", you could not see your hand in front of your face... so to speak.
That may be right but luckly we don't have to answer that particular question because it is not possible to travel at the speed of light. That is a dodge - but an acceptable dodge.
If you were moving half the speed of light with your headlights on the light would travel half as far "ahead of you" per (insert omnipotent timeframe here) as it would if you were "sitting still" for the same duration. If you were travelling at .5c with a light source behind you the light would be 450,000 km away from you after 1 second.
You are really going to hate this response! All I can say is every observer will measure the speed of light at c. Every observer without exception. So what does this mean in this case:
For the person on the ship traveling at .5c - the speed of light would be 300,000 km/s, so the photons from the headlights would travel 300,000 km after 1 sec from your perspective, and the photons from your tail lights would travel behind you 300,000 km after 1 sec from your perspective.
An observer that was stationary relative to your space ship would observe the light emitted from the front of the ship would only be 150,000 km from the ship after 1 sec and the light emitted behind the ship would be 450,000 km from the ship after 1 sec. The reason for this is because the 'stationary' observer would see the ship moving and he would also measure the speed of light at c.
Look at it this way if a space ship flew by at 99.99999999999999%c and you could somehow see the ship and a photon moving past the ship it would look like the photon would just barely catchup to the space ship and then pass it at like 25 kph and then ever so slowly move ahead of the ship. If you were on the ship watching that photon pass you it would be moving (all together now) at 300,000 km/s.
Now the reason this is important is because there is no 'omnipotent timeframe'. That is to say there is no possible way to say this is 'really' the rate of time passing. That is because there is no way to say what is stationary or moving. Motion and therefore the passage of time is relative.
That was the whole point actually of the Michelson–Morley experiment, they wanted to see how fast we were moving through the aether by measuring the differences in the speed of light as we went towards and way from the sun. Einstein correctly realized that since this experiment and others always returned the same speed of light that it was constant. This invariant speed led to realization that there is no way to define absolute motion or even absolute time.
It is all relative.