Do I need an adjective or adverb here? I usually use fast so that there is no visible difference but I hear people say things like "Lets do this real quick" all kinds of colloquial stuff so I am not certain whether "really quickly" is.
In your example "quick" modifies the verb "got" so the adverb "quickly" is correct.
However, that being said, "quick" is often used colloquially as an adverb. Since you use "got" instead of "became", "He got angry real quick" is natural colloquial speech.
If you want to write the same thing more "formally", you'd instead say something like, "He quickly became angry."