Modern History and Revolution
Samuel Eliot Morison, The Oxford History of the American People 
Howard Zinn, A Peoples History of the United States
Zinns book is an important corrective to the usual histories and textbooks, but thats all it is. If you rely only on his book, you will have a narrow and distorted idea of American history. I suggest that you first read some mainstream work Morisons is probably as good as any then read Zinn.
The Federalist Papers [1787-1788]
A series of articles by Alexander Hamilton, James Madison, and John Jay, arguing for the adoption of the new American Constitution. They are among the most masterful political polemics ever written, and as such well worth reading, though the Constitution they argued for served in many respects to reinforce vested political and economic interests. The first half of the book is the most important; the later articles (numbers 52-83) concern details that are of less general interest. For the opposing arguments, see Cecelia Kenyons The Antifederalists or Jackson Turner Mains The Antifederalists or, if you want to go explore the matter in detail, The Debate on the Constitution (Library of America, 2 vols.).
Henry Adams, History of the United States During the Administrations of
Thomas Jefferson and James Madison 
A lucid, leisurely, and elegant account of the early years of the new republic (1800-1817). If its too long for you (it takes up two hefty Library of America volumes), try the first six introductory chapters, sometimes published separately under the title The United States in 1800. Garry Wills’s Henry Adams and the Making of America shows that the History has a quite different “lesson” than is usually thought, and he also makes a good case that it is one of the prose masterpieces of the nineteenth century.
Alexis de Tocqueville, Democracy in America
Possibly the best general study ever written about the United States. Many of Tocquevilles insights into American culture, character, and social organization remain valid.
Herbert Apthecker, American Negro Slave Revolts
The sheer number of revolts (approximately 250) documented in this pioneering study refuted previous contentions that the slaves were resigned to their lot.
Frederick Douglass, Narrative of the Life of Frederick Douglass
Autobiography of the escaped slave who became a leading Abolitionist spokesman. The Narrative is quite short. If you want more, there is a later and longer autobiography entitled The Life and Times of Frederick Douglass.
W.E.B. Du Bois, The Souls of Black Folk
Militant without ceasing to be humanistic, Du Bois was the most important African-American spokesman of the early twentieth century. This is usually considered his best work, but he wrote many others that are essential for those interested in African-American history.
Louis Adamic, Dynamite: The Story of Class Violence in America
Classic account of the violent class struggles in America that have been left out of the usual histories and textbooks.
Jeremy Brecher, Strike!
Similar to Adamics book, but includes more recent history. The new 2014 edition brings the story up to the present.
Harvey OConnor, Revolution in Seattle
Good account of the 1919 Seattle General Strike, including the struggles by the IWW and others that led up to it.
Sidney Fine, Sit-Down: The General Motors Strike of 1936-1937
Detailed account of the important and innovative strike in Flint, Michigan, in which 1200 auto workers occupied their factory for six weeks.
Robert Caro, The Years of Lyndon Johnson (4 volumes so
This immense work-in-progress is the best political biography I’ve ever read. Johnson is certainly not one of my favorite people, but I think Caro is right in seeing him as one of the key figures of the twentieth century, so that in reading about his life you learn quite a bit about the workings of American political power during the period from the Depression through the sixties.
The first four volumes bring the story up to early 1964, when, following the assassination of John F. Kennedy, Johnson uses his political skills to ram through the Civil Rights Act despite the stubborn resistance of the Southern segregationists with whom he had previously been allied. (He needed to do this to ward off an otherwise likely primary challenge from his enemy Bobby Kennedy.) Those first four volumes come to 3300 pages! That may seem like a lot, but whole story is so superbly organized and narrated I could hardly put it down. The fifth and final volume will cover the Vietnam War and other events of the late sixties, but we will apparently have to wait a few more years for its appearance. Caro is an extremely thorough researcher (the previous volumes have taken him about ten years each), and he plans to visit Vietnam before completing his work. Meanwhile, if you want to try just one volume, I suggest Volume 3: Master of the Senate (1948-1960), where Johnson is really in his element and his astonishing political genius is at its most evident.
John Howard Griffin, Black Like Me
In 1959 the author, a white man, disguised himself as a black and traveled through the American South. His best-selling account of the shockingly different treatment he got due to a mere change in skin color was of considerable significance in generating white support of civil rights struggles.
[Rexroth essay on the civil rights movement]
Malcolm X, Autobiography
Malcolm Xs political ideas seem to me to be much overrated, but his autobiography is very engaging.
James Carr, Bad: The Autobiography of James Carr
Tough, no-illusions story of a young black mans experiences as criminal, a prisoner, and a Black Panther. He was in the process of developing a more antiauthoritarian radical perspective when he was assassinated by two ex-members of the Panthers.
I have not read a single book that seems to convey a very accurate idea of the sixties counterculture or New Left movement, much less of the sixties as a whole. In each case certain aspects, or even merely certain personalities, are played up as much more important than they were and many other aspects are distorted or neglected. I suppose this is inevitable with such an immense topic, where so much was going on in so many contradictory directions. Here are some books that may give you a taste of one or another aspect:
Charles Perry, The Haight-Ashbury:
A gossipy but generally competent account of the Haight scene during its heyday as capital of the hip counterculture (1965-1967).
[On the Poverty of Hip Life]
Art Kleps, Millbrook: The True Story of the Early Years of the Psychedelic
An amusing insiders account, centered on the mansion in upstate New York where Timothy Leary, Kleps and others were headquartered during the early sixties. There are many other books on the psychedelic movement, but they tend to be superficial and sensationalistic. Klepss narration is certainly trippy enough, but he knows what he’s talking about and his sense of humor and irony helps keep it down to earth.
Tom Wolfe, The Electric Kool-Aid Acid Test
I dislike Wolfes glib, trendy style of journalism, but this particular book covers some notable events in the sixties psychedelic scene in considerable detail.
Emmett Grogan, Ringolevio
Autobiography of one of the founders of the San Francisco Diggers. Its an interesting and sometimes exciting story, but it should not be taken too seriously: Grogans grandiose sense of his own importance gives a rather distorted impression of what things were really like.
David Lance Goines, The Free Speech Movement
An excellent and comprehensive account, by far the best book on the Berkeley FSM. The author was one of the leading participants, but he rounds out his personal narrative with generous excerpts from other participants takes on the events.
[Remarks on the FSM]
Michael Rossman, The Wedding Within the War
This chronicle of the movement, 1958-1970 by an FSM participant well conveys some of the personal and subjective aspects of the New Left scene. A later book by the same author, New Age Blues: On the Politics of Consciousness, though interesting in some respects, is in my opinion too uncritically accepting of the existence of supposed paranormal phenomena.
Todd Gitlin, The Sixties: Years of Hope, Days of Rage
Good history of the New Left movement by one of the early SDS leaders. My review of another Gitlin book, The Whole World Is Watching: Mass Media in the Making and Unmaking of the New Left, is online here.
[Situationist critique of the sixties student movement]
[Critique of the New Left]
Peter Stansill and David Mairowitz (ed.),
BAMN: Outlaw Manifestoes and
Ephemera, 1965–1970 
A confused and eclectic collection, but it gives some idea of some of the more radical agitational currents of the time, in Europe as well as America (Situationists, Provos, Diggers, Yippies, Motherfuckers, etc.).
Section from Gateway to the Vast Realms (Ken Knabb, 2004).