Lorem ipsum dolor sit amet, consectetur adipiscing elit. The circle of fifths is usually drawn as wall art, but it is really a lookup table: spin clockwise and each step takes you to the dominant; spin the other way and you land on the subdominant. The hop from C to G is one click clockwise.
Reading the wheel
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Each position on the circle is a key, and its distance from the top tells you how many sharps or flats the key signature carries. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.
Closely related keys
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. The keys immediately beside any tonic differ from it by a single accidental, which is why a move to the F feels gentle while a leap across the wheel feels like a hard cut. Excepteur sint occaecat cupidatat non proident.
Planning a modulation
Sunt in culpa qui officia deserunt mollit anim id est laborum. To leave the home key without jarring the listener, pivot on a chord the two keys share, then let the new V confirm the destination. Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam.
From wheel to voices
Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores. The circle tells you which chords are next door; Voice Leading in Four Parts tells you how to get there one voice at a time. For the canonical diagram and its history, see the Wikipedia article on the circle of fifths. Neque porro quisquam est qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit.