I really should get back to work, but I wanted to say something more about the circle of fourths/fifths:

Going around the circle of fourths part of the way is a common idiom in many popular songs (and other kinds of tonal music). In the key of C, perhaps C F Bb Eb and then maybe E7 to Am, for example. The C F Bb Eb is a short trip around part of the circle of fourths. C and F are in the key of C (or Am), with Bb, we go out of C (because of the Bb itself), Eb is even further out (because of the Eb), then a slight shift chromatically to E7, which is outside of C but the dominant of Am and we resolve to Am. If we want to, we can just continue in C, since C and Am are very nearly the same key.

Eb maj. to E7 is this:
Eb G Bb --> E G# B D

Leaving the specific voicing aside, each note only moves a single half-step: Eb --> E and D, G --> G# and Bb to B. This sounds good to most people's ears, if not overdone, of course.

It can be made more interesting by using a seventh chord on the Eb. For Eb maj. 7, there would be an additional D, which would just stay put. One can do interesting things by using different kinds of seventh chords. What really makes things interesting is the voice leading, which means paying attention to the movement of the individual voices. There's no end to it.

Jerome Kern's "All the Things You Are" is a good example of a song with passages using the circle of fourths. (I can't remember who wrote the lyrics. Oscar Hammerstein II, perhaps?)

It's worth noticing that Bb and Eb are notes in the key of Cm and are also in the C blues scale. One way of putting this is that one is "using the expanded tonal space". It's quite common to alternate between the major and minor of a given key, or to mix the major and minor of a given key. This happens when playing blues. It sounds good to most people because sticking to just one key can sound a bit dull. The next chord in the circle of fourths would be Ab and Ab is also in Cm. This can't be a coincidence.

An interesting thing about the circle of fifths is that one can continually use it to continually resolve to the next chord in the circle. For example, G is the dominant of C, so I could play G7 - C maj. 7. But C is the dominant of F, so I could play G7 - C7 - F maj. 7. Or G7 - C maj. 7 - C7 - F maj. 7 - F7 - Bb maj. 7, etc. Or just G7 C7 F7 Bb7 Eb7 Ab7, etc. A single and fast way of getting into another key, for a shorter or longer time. Other seventh chords can be used, too, but dominants tend to make one feel that they should be resolved.

Learning to recognize these idioms is also very helpful when trying to work out songs by ear.