For nonfiction, I would recommend all Chomsky and Zinn as high priority. If you haven't read the People's History of the United States before, you should definitely pick it up. All Chomsky is good, but can be quite boring and tedious. I must confess that I have never picked up one of his books and read it cover to cover in a short amount of time. I always read some, read another book, and then come back. His books cause an information overload, but it is all good information.
Both of those authors are anarchists, but their books are about a lot of different things.
If you want the basics of anarchism, you should read Peter Kropotkin's Mutual Aid. Also, Bakunin is good. I haven't read any of his books, but I have read his essays.
A personal favorite of mine is Shibboleth: My Revolting Life by Penny Rimbaud of Crass. It is basically the story of his life, but anarchy is an underlying concept in the book. To see how some anarchist thought actually played out within the punk community of the time, there is no better read.