Mathematically "Precise" Explanation of Group Theory

In high school, I messed around a lot with Non-Euclidean Geometry. I liked it so much, that I embarked on a university career with specializations in Math & Philosophy. However, in math, I couldn't get my head around Group Theory. I knew it wasn't all that difficult, but I just couldn't grasp it. It was, in part, responsible for my abandoning that career and taking the more prosaic route of joining the work force.

Now, 35+ years later, I find myself reading two biographical books that are heavily into both Group Theory and non-Euclidean Geometry. In addition, a friend just forwarded me three articles on interactions between art and the sciences, the third of which was about Non-Euclidean Geometry and Art (any M. C. Escher fans out there?).

After all these years, my appetite has again been whetted to delve into these esoteric subjects. But will I be able to get my head around them this time? Who knows? But the fun will be in trying.

I am already encouraged by a quotation from "The Equation That Couldn't Be Solved", one of the books I am reading. The author (Mario Livio) quotes James R. Newman from his monumental compilation "The World of Mathematics":

"The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing."

This whole entry was just to share that gem of mathematical precision!!

It really sounds like an explanation by Douglas Adams, doesn't it? The funny thing is, I think I understand it now!