What is the correct word to use here and why:
I will get there quicker [than you]
I will get there faster [than you]
There must be similar adverbs for “slower”.
The definition of the two words makes them synonymous in virtually all cases. However, they do have slightly different connotations that lead to preference in usage. I generally think of something as “fast” if it can achieve a high speed. I think of something as “quick” if it responds rapidly to input. This generally leads to preference of one word over the other in context; “fast” is used in context of speed, while “quick” is used in context of time. So, you would travel fast to get somewhere quickly.
In the same vein, “quick” is used to describe the quality of an action that is short and powerful, e.g. a “quick head-fake”. “Fast” is generally used to describe actions that are more sustained, e.g. “a fast sprint down the field”.
None of this is concrete; you hear of someone, say in a race, having “the fastest time” much more often than “the quickest time”. Clearly, the context is time, not speed, but use of “fast” is preferred anyway.