The Epic of Gilgamesh (Anonymous) - This is my favorite book; it also happens to be the world's oldest know work of literature, so I figure it's a great place to start. It's the archetypal buddy story, with adventure, tragedy, homoerotic overtones, and in the end, a knock-your-socks-off philosophical enlightenment. Best of all, it's only about 100 pages long. Try to find the Penguin Classics edition - or some other prose translation; they're much more readable than the verse translations.
Some other ancient literature:
The Odyssey (Homer) - I agree wholeheartedly with this choice. Action packed, and the basis for many, many cultural references and later works of literature (you'll finally understand the movie O Brother Where Art Thou!)
The Histories (Herodotus) - It's long and sometimes tedious, but this history of the rise of the Persian empire and the eventual war with the Greeks contains some great stories, including the basis for the movie 300.
Now, let's jump ahead to the 20th century.
Ask the Dust (John Fante) - This is a very honestly written book about a young guy (20 in the book, he was 25 when he wrote it) struggling to find himself and achieve artistic success, mostly doing a crappy job of it, in Los Angeles during the Great Depression. The language is plain and straightforward, making it a much more modern read than most literature of the era.
The Grapes of Wrath (John Steinbeck) - An epic tale of Dust Bowl farmers pushed off their land by the thousands, moving to California in search of the American Dream and finding low wages, inhuman conditions, and starvation working as fruit pickers in the Central Valley. For great literature dealing with raw human emotions, this is about as good as it gets.
The Big Sleep (Raymond Chandler) - You can't live in the world of 20th centry literature without taking on the genre of the noir hard-boiled detective story. Chandler's writing crackles off the page, even though his plots are fairly corny by post-modern standards.
On The Road (Jack Kerouac) - This novel spawned an entire subculture - the so-called "beatniks". It's the early 50s, and Kerouac and his friends bum around the country, smoke weed, hang out in dank jazz clubs listening to Negroes blowing saxophones. Written in a stream-of-consciousness style which can get tiresome and childish, but if you like this one, check out his later books The Subterraneans and Big Sur - they're much better written and have a bit more story going on.
The Electric Kool-Aid Acid Test (Tome Wolfe) - A nonfiction work about the LSD subculture of the 60s. Ken Kesey (he wrote One Flew Over the Cuckoo's Nest) and his band of "Merry Pranksters" travel around the country in a broken-down old bus, dropping acid, wearing outlandish clothes and generally shaking things up.