Light travels at the speed of light regardless of your speed.
- Light coming towards you would be going at the speed of light relative to you.
- Light going in the same direction as you would be going at the speed of light relative to you.
- That same light would also be going at the speed of light relative to a stationary observer that you both passed.
This is a fundamental principle of special relativity. The constant and the fastest speed is $c$ which is the speed of light in a vacuum. In other media, it may go slower.
For example, Cerenkov radiation occurs when particles go faster than the speed of light in the atmosphere. The particles come in at speeds that are very close to the speed of light in a vacuum but faster than light in the atmosphere. As the hit the atmosphere they slow down but for a brief while they are going faster than the speed of light in the atmosphere. Actually analogous to breaking the sound barrier.
Anyway, to return to your question, this may seem impossible. This is because you have familiarity with much slower objects and have built an intuition that, for example, if you were travelling at the speed of a bullet:
- A bullet coming towards you would be going at twice the speed of the bullet relative to you.
- A bullet going in the same direction would be stationary relative to you.
- That same bullet would be going at the speed of the bullet relative to a stationary observer.
This intuition is correct only for non-relativistic speeds. The relativity equations also predict this for slower speeds and reduce to the Newtonian equations that yield this intuition.
At higher (relativistic) speeds what changes is time. Time is not the same for all observers and this is how the puzzle is resolved.