Quikly

You will hear people say all three interchangeably. Don't have a grammatical rule at my disposal but all 3 are commonly used from my experience.

If you are using these words as Adverbs (adjective modifying a verb), they are all applicable - BUT "quickly" is the most formal, then "fast", then "quick" (which is extremely informal and not used much by most people). In writing you'd only use the strictly correct adverb "quickly". Of course, if you were modifying a noun it would be completely different. Usain Bolt is Fast. You can't say "Usain Bolt is Quickly". But in your case you are modifying "changes" (a verb) so Quickly is a good choice and is a bit formal. You wouldn't hear a rapper saying Quickly here, and you'd never hear a professor say Quick in this scenario either - but all three could be used. Or you could say "The weather changes Frequently in many places in the US" - which might be what you meant to say anyway :).