Oh boy, what a question. Let me leap into this with both feet.
Start with the three-volume The Americans by Daniel J. Boorstin. Not your conventional American history, it is instead a series of chapter-long essays on the important factors that shaped the American experience. Its greatest value is that it is one of the few books to get away from the Great Man theory of writing history. In its place, Boorstin shows that invention, infrastructure, innovation, and the Industrial Revolution were a thousand times more important in creating the America we see today.
If you can read no other books about American history I would still recommend this one first. And Boorstin, the former Librarian of Congress, is one of the world’s great writers of history. All his books are good and worth reading. His three volumes, The Discoverers, The Creators, and The Seekers, do for world history what The Americans does for our country.
Now to go through our history in more or less chronological order. I’m not going to mention any biographies, even group biographies. In the first place, there are millions of them. And most of the ones coming out these days are pretty good. We are in a new golden age of historical writing, clear, well-written, well-researched, and aimed at the general reader rather than fellow historians. You have to go back to the 40s and 50s for another such age. By all means read them. It’s just that you don’t need me for that.
For colonial history, you can start with The Bold and Magnificent Dream: America’s Founding Years 1492-1815. The marvelous Civil War historian Bruce Catton wrote it with his historian son William B. (Catton’s Civil War books were the highlight of the Centennial in the early 1960s and are still well worth reading.)
If you really want to get to the heart of the writing of the Declaration of Independence, forget books and rent the film (or even better, see the play) of 1776. I’m serious. No book that I know of does this miracle of compaction in setting out the issues and the characters and sectional rivalries as well as this does - and it does it with the best witty conversation you’ve ever heard and lots of good songs.
The Revolution. Rather than sort though war books, I’ll make a more or less blanket statement that you should find all the various books that American Heritage has put out on American wars. Some of the them are old, although they get revised regularly, but they all use top historians who were told to write for reading, not for research (or edited that way).
There are two essential books on the writing of the Constitution: Miracle at Philadelphia by Catherine Drinker Bowen and Decision at Philadelphia by Christopher and James Lincoln Collier. The first looks more at the personalities; the second more at the issues. They complement each other nicely.
Stephen E. Ambrose was simply the best writer among modern historians. His books are, pardon the phrase, unputdownable. He slips in here in chronological order with Undaunted Courage: Meriwether Lewis, Thomas Jefferson, and the Opening of the American West, but you should also read Nothing Like It In the World: The Men Who Built the Transcontinental Railroad 1863-1869 and Crazy Horse and Custer: The Parallel Lives of Two American Warriors.
I have a problem here because the next set of years is unbelievably essential for understanding what comes after, but I don’t have any good popular histories to recommend. One historical novel that does tackle the subject of slavery and abolitionism in those decades is the monumental Cloudsplitter, by Russell Banks, a huge book that studies John Brown’s life through the eyes of his son. I know of nothing better on daily life and passions in the US in that era.
For the Civil War, there are two routes to take. One is the overall history, than which in one volume there is no better example than Battle Cry of Freedom: The Civil War Era, by James M. McPherson. (Christopher Ward’s book supplementing Ken Burns’ Civil War documentary is surprisingly excellent and of course supremely well illustrated.) Shelby Foote’s three-volume Civil War is even more outstanding, but it’s friggin’ huge. (How huge? They excerpted just the piece on the battle of Gettysburg and issued it as a 250-page book.)
Or you can poke around the war from the sides. I won’t go through the gazillion books on individual battles, but I must mention a classic work of history: Reveille in Washington, 1860 - 1865, by Margaret Leech. It makes the city come alive and gives the feel of what it meant to live through that war at close range for everyone from Lincoln to the prostitutes the soldiers visited.
Nobody actually likes reading about Reconstruction, a horrible period of history that depresses everyone who comes near it, but the standard is Reconstruction: America’s Unfinished Revolution, 1863-1877, by Eric Foner. There’s even an abridged version now available.
Cowboys. The West. How can you read American history and not read about cowboys? So the place to start is Cowboy Culture: A Saga of Five Centuries by the noted western historian David Drury. Two big illustrated books with text good enough to read for itself are The Wild West, another documentary spin-off and The Native Americans: An Illustrated History.
Post reconstruction came the most crooked period in American history and boy is that saying a lot. Ray Ginger’s 1965 Age of Excess: The United States from 1877 to 1914, one of the last great narrative histories of its era captures these decades as well as anyone. Then you can go on to read Lincoln Steffens’ early century classic The Shame of the Cities to see what the result of unchecked greed can do. (I would write “unchecked capitalism” but some Libertarian would come along to diss me, not knowing what a champion of capitalism I am.)
Walter Lord was already mentioned, but I like Frederick Lewis Allen. His 1930 Only Yesterday is a remarkable thing: a good historical book about a period that had just ended, the 1920s. (He extends this look at both ends in The Big Change: American Transforms Itself 1900-1950.) William L. O’Neill manages somewhat the same feat for the 1960s with his 1971 Coming Apart, though it is less popularly written than Allen.
Fat and sassy is William Manchester’s The Glory and the Dream: A Narrative History of America 1932-1972, which captures the depression and the war quite well, but it weaker on the later years.
Nixon led to some wonderful books slagging him, from Fear and Loathing on the Campaign Trail 72 by the inimitable Hunter S. Thompson to that Watergate must: All the President’s Men, by Carl Bernstein and Bob Woodward.
I think I’ll stop there because a) it’s already too much and b) it’s the exception rather than the rule to find good histories about recent years. Anyway, this is just a tiny sampling of the thousands of wonderful books on history and a biased sampling at that since I confined myself to my own shelves.
American history is wonderful, amazing, astounding, horrifying, depressing, masterful, and a sight to behold. Dive in deep.