If I'm in a spaceship travelling at light speed, what happens when I run from the back to the front?

It's the theory of relativity.

You are moving at walking speed relative to the ship.

To anybody outside at (what you would consider) "rest", you would still be moving at light speed.

Although, I don't believe they'd be able to tell you apart from the spaceship itself because all of the photons bouncing off of your ship for the entire journey are hitting his eyes at exactly the same time, so it would probably just be one giant flash of light.

Anyways, you won't be going faster than the speed of light, but if you approach a viewer at the right angle you could APPEAR to move faster than the speed of light.

Umm...I'm not really sure which would be the correct answer!  Considering the law of Inertia, nothing should happen if you are in the spaceship, moving right along with it...however I doubt the example of a moving plane applies to the lightspeed spaceship.  So...I'd say if it's already established it's motion, you would be normal, however if it's still building momentum you'd fall backwards...just my novice opinion XD

My guess would be that you would still be at rest, relative to the observers outside.

Because of time dilation (an axiom of special relativity), 1 second outside of your spaceship will be 0 seconds for you, which means that time is "frozen" . You won't be able to notice this because you are in an inertial referance frame, so you will be walking at walk speed and your watch will be ticking at the same rate as if you were sitting still.

Ok, to answer your question, running to the front of your spaceship will take forever, literally. This is all due to the time dilation.

At relativistic velocities, you can't just add speeds like you can at everyday speeds.
This is because strange things happen to both distance and time at relativistic speeds, so speeds, and how you add them,  get decidedly strange.

This is why a beam of light from a spaceship moving at .99C will be seen as moving at C by all observers. Nobody will see it moving at 1.99C.