It is a common misconception that light speed itself changes in a medium, what changes is the rate of PROPAGATION. It is not the photon that is slowed, it is only the propagation of information that is slowed (the photon is absorbed and re-emitted).
One way to understand the implication of relativity is to understand that spacetime is one thing and you have a constant velocity vector through this spacetime. And the absolute value of every velocity vector, in spacetime, is constant = 1.
So everything is always moving through spacetime at "the speed of light". But what happens is the direction of the velocity vector can be tilted more towards space dimensions or more in the time dimension.
An object at "rest" in space moves through time at a unit rate of 1 (consider units where c = 1). As the object starts to move through space the velocity vector is tilted slightly causing the object to move less through time and more through space.
There are other effects as well but that's a good summary that can help people understand the effects in a more natural way. It also explains why you can't go faster than light-you already are moving at 'lightspeed' through spacetime.