I find the circle of fifths a helpful tool in learning the common relationships of key signatures and middle of song key changes (modulations) by writing out the Major scales when I use this method. It also helped with understanding "#'s vs. b's" (sharp's vs. flat's). The Circle of Forths is common for using flats starting in the opposite direction from the fifths (some diagrams show fifths clockwise, others counter-clockwise). Either way the fifth of C is G and the forth of C is F.
The method that hit home was looking at the theory in a straight line, like this. It starts and ends at the key of C. Then take the fifth of C and start a G Major scale.
C-Major scale (up 2 octives)
C-D-E-F-G-A-B-C-D-E-F -G-A-B-C
-G-A-B-C-D-E-F#-G-A-B-C-D
D-E-F#-G-A-B-C#-D
And so on..........................................
When you complete B you will have 5 #'s and then F# will make 6 #'s.
From that point start over again with C but this time use the forth note which is F and do the F Major scale. It will contain 1 b note. Continue the forths until you reach Gb which will make 6 b's (the same note as F#) and the circle will be complete. When making a modulation (change in key) the more common notes shared between the keys the smoother the transition.