Does the equivalence principle imply that light must move slower when moving away from a massive object? Thought experiment: Elevator going up at an extreme acceleration, pulse of light bouncing up, and down between mirrors on the floor, and the ceiling. Won't it take light longer to travel from the floor to the ceiling, than from the ceiling to the floor? If so,then based on the Equivalence Principle, doesn't this mean that light will move slower from floor to ceiling in an Equivalent gravitational field? 
 A: Your reading of the thought experiment is not correct.  
The equivalence principle implies that locally the laws of physics are described by special relativity, hence locally the speed of light is $c$.  
The thought experiment means the ceiling (receiver) is moving away from the floor (source) and thus measures a lower frequency of the light. Conversely the floor is measuring a higher frequency. It is the relativistic Doppler effect in special relativity.  
Applying the equivalence to a gravitational field, in the former that would show as the gravitational redshift (light moving away from a massive body) and in the latter as the gravitational blueshift (light approaching a massive body).
A: Light is ALWAYS traveling in  the same speed of $c$ in ALL reference frames. This was confirmed by various experiments such as Michelson & Morley experiment and the others.
The only thing that changes - is the light frequency,- if light looses energy somehow then it's frequency is red-shifted, but speed is the same $c$. Of course if photon is traveling in vacuum. If photon is traveling in medium with refractive index of $n > 1$, then phase velocity of light is $v < c$. The only reasonable explanation if light speed is the same everywhere - is that time flow changes and is dependent on reference frame (this was solved by Einstein). I suggest you first to read about special relativity, because Einstein has developed it first. Then study general relativity because it is much much more complex that comes after.
