Many English learners use the word “fastly” as an adverbial form of “fast”, which seems quite logical because this is the way adverbs are usually formed. If something is slow, we say that it “moves slowly”, and such usage is completely correct. Nevertheless, languages develop in a way that is not always logical, and the situation of “fast” vs. “fastly” falls exactly into this category.
The fact is that there is no such word as “fastly”. “Fast” is already both an adjective and an adverb, as in
It doesn’t matter whether “fast” refers to the speed of movement or the rate at which something is being done; it’s always just “fast”:
The word “quickly”, used as an adverb, is synonymous with “fast”, but usually refers to the time an action takes rather than to the speed of movement. For example, you can say
which means that you want the other person to come soon; you don’t really care how fast he or she is moving. Nevertheless, “quickly” can also refer to the actual speed (as in “he runs quickly”), but such usage is less common.
There is one situation in which it is obligatory to use “quickly” (or “swiftly” or a related adverb ending with “-ly”)—if an adverb precedes the verb it modifies: