Most key changes involve chromatic lines or chords:

A change from Fmaj to Emaj, for example, would go something like this

F-F#-G-G#-A-Bb-B-C-C#-D-Eb-E-E-E
in the melody (though it certainly doesn't have to be a straight shot like that.

And the harmony might be something like, hmm...

Fmajor7 (3rd inversion) : Gminmajor7 (root position) : Gminmaj7 (third inversion)

etc etc etc

Or you could bring in secondary dominants, i.e. V of (in this case) Vii