Marvel's Civil War was my introduction to modern-day comics, and to comics period. I still consider Civil War to be a great jumping-on point for curious readers. A lot of the relationships you see between characters even today can be traced back to Civil War.
If you like Joss Whedon's snappy-paced, dramatic and funny direction and impression on the recent Marvel Studios movies and Cinematic Universe, then you could probably enjoy the contemporary writing style of Brian Michael Bendis in Civil War and its associated collections. In fact, once you begin to pick up and read more contemporary comics, you'll really begin to see what a major influence they're having on the movies and vice versa.
All 7 issues of Civil War are on sale for 99 cents each at the Marvel Comics Digital Store.
http://comicstore.marvel.com/Civil-War-Sale/comics-collection/1477Civil War was the biggest crossover event in years, so there are other books surrounding it. You don't have to read them (really), but if you want to keep reading, those books are also listed for 99 cents each, this weekend only. If you're wondering which to pick, just pick the one that's about the character or events you're most drawn to. If you decide you want to read them all in suggested order, Google for
Marvel Comics Civil War reading chronology and pick a guide that you like or just use
this one.