I did not initially understand the first isomorphism theorem.  It was not until my first year of graduate school, when my wife wanted me to teach her some group theory, that I really hit on a natural sequence of ideas which made the first isomorphism theorem "click" for me.  I talk about it here:

http://mathoverflow.net/questions/41955/does-any-textbook-take-this-approach-to-the-isomorphism-theorems