At the speed of light, it doesn't. Seriously, time ceases to pass for anything travelling at C.

For anything moving through spacetime, the passage of time is slowed. The magnitude of this slowing (the "time dilation factor") increases as velocity as a fraction of C increases. Yes; this means that right now you are technically experiencing Special-Relativistic time dilation because of Earth's motion through spacetime (and have been for your entire life, so it's all you know).

This is not a linear relationship; the time dilation factor is very small until you achieve a significant fraction of C. It doesn't reach 2 (so time is passing at 1/2 the base rate) until something like 0.85C. As your velocity approaches C, the time dilation factor tends to infinity (and the passage of time slows to an arbitrarily slow rate).

Technically speaking, nothing with mass can travel through spacetime at C, only at arbitrarily-high fractions of C (which also takes inconceivable quantities of energy, but that's a separate discussion entirely). The key words there are "through spacetime", though. If you can find a way to bypass spacetime or modify it in some way so that you can cover a longer distance faster without actually going faster *through* that spacetime, then you can avoid the Special Relativistic effects. That includes the time dilation *and* the prohibitive energy costs.