The circle of fifths is also a common way to make a key change work. You play through chords one fifth away from each other to reach your desired key. Normally you only play through a couple.

For example, say you are in the key of G and you want to get to another key. Well, the fifth of G is D and you could simply play the G and then the D and be in a new key. And off you go in the key of D now. Or you could go a step further: G D A (A is a fifth away from D). Now you've gone from the key of G to A.

Relative minors can make the transitions a bit smoother, too, because they change the sound of things without actually taking you away from your target key. For example, you could go G Em D Bm A or G Bm D F#m A and they work quite well.