A visitor to Play Ukulele By Ear, Sven, brought to my attention a website called Hooktheory, that breaks down the chord progressions to popular songs and puts them into number form. The number system is the best way to begin to understand the underlying structure of musical compostions.
As you can see in the screen shot example of Time Is On My Side by The Rolling Stones, the structure is your basic I-IV-V (Key of C: C-F-G) chord progression. That is followed by a vi-V (a minor-G), vi-V/V (the V/V is a 5 of 5 chord which means it is the 5th of the 5th, in this case a 5th above G is D or II-D Major).
The colored squares at the top of the example give you the melody notes to the song, and you can even change the key the song is played back in! The only drawback is that the entire song is not displayed (probably due to copyright concerns). Nonetheless, Hooktheory is a great site to start to learn how to start thinking in numbers as opposed to letters when learning new songs.