In the last 100 years, the music output from American culture has become a dense and intricate web of interesting history, with new genres created and adored practically every few years, and musical styles splitting, forming, coalescing and ultimately mixing back with the styles they split from in the first place. It's all a very evolutionary process; the DNA of musical styles mutated by individual artists, their survival decided by record companies and ultimately listeners.
But the origins of American music can be traced back to relative simplicity. Essentially, if you learn nothing else, remember that almost all American music stems from the mixing of European folk music (mostly Irish) and African folk music. The Irish, Scottish, English, French and Spanish immigrants, many living in the Appalachian mountains, started hanging out with African former slaves, who's polyrhythmic spiritual music was already gaining popularity after the civil war. The Civil War itself brought many whites and blacks together simply out of necessity; soldiers' fighting together shared their music with each other. Modern bluegrass is the closest reflection of this ancient (by American standards) combination of styles.
Negro Christian Spiritual hymns gained popularity in the late 1800's, which were essentially old European hymns sung in an African call-and-response polyrhythmic style. Then near the end of the century, a peculiar African American dance gained popularity; an over-the-top parody of ballroom dancing called the Cakewalk, often accompanied by goofy costumes. The intense popularity of this led to ragtime, which then evolved into jazz, thanks mostly to African American marching and, bizarrely, funeral bands from New Orleans. Jazz, of course, became possibly the single most influential change in America music and led to blues and rock n' roll. You probably know the rest.
It is interesting to note that if it weren't for African Americans, the United States would probably still be listening to John-Philips Sousa and Irish-Appalachian jigs.
This is all a gross oversimplification of the full history of American music. I didn't even go into Native American music, Cajun, Latin-American, or any of the several other cultural styles that have had their effect on our culture. The amount of literature related to this subject seems near infinite; the depth at which you can research any particular sub-topic is really only limited by your own conviction. Now that you have this tiny bit of information, you can strike out on your own and maybe hear some new stuff on the way.