Learn how the black keys map to sharp and flat notes, then learn the cycle of fifths. (There are also some "tricks" you can use to understand how sharp and flat key signatures relate to their keys: in a "sharp" key signature, the last sharp sign matches the leading tone for the corresponding major key (e.g. G major only has F# as a key signature sign); in a "flat" key signature the last flat sign matches the fourth, and the next to last matches the actual key. F major has a single flat sign at Bb; Bb major has two, at Bb and Eb.) Once you have internalized how all of these relate, it really becomes trivial.