The two must read books of American history are
Empire of Liberty (1789-1815), Gordon Wood.
Rise of American Democracy (1815-1860), Sean Wilentz.
Both are fine stand-alone, synthetic treatments of their eras- coffee table books, in effect. They are long, and with lots of material, but they are explicitly designed as introductions for non-specialists, but are subtle and important enough that they are go-to references for professionals in the field.
More specific monographs
Aggressive Nationalism- Richard Ellis. Constitutional conflict in early 1800s America
Free Soil, Free Labor, Free Men- Eric Foner- the antebellum political era and its conflicts.
Constitution Besieged- Howard Gillman- evolution of political thought from 1880s to 1920s
And, since you just saw Lincoln- from another thread
Read James Oakes's The Radical and the Republican; a subtle, challenging, but sympathetic account of Lincoln [and Frederick Douglass] with a particular focus on his involvement with party politics [what this film is mostly about]. This is the book I'm recommending to everyone interested in this film [and not just because the historian who wrote it is a great guy with whom I studied.]
I'd also second Tocqueville, but only AFTER you read the first two books on the list. 1871 is right that it's not the best starting point, but it's the best book on America. However, it requires the historical background that Wood and Wilentz will give you.
EDIT:
It's fun and edgy to post Zinn, Lies My Teacher Taught Me, etc., but the professional historians are all shying away from those sorts of works now and returning to something closer to orthodox history. I can tell you that all of the serious work we read in my history PhD program was far closer to old-school history- you know, the "Lies My Teacher Taught Me"- than anything out of Zinn and friends. And none of my history professors were anything close to Republicans, for the record. Foner himself is a raving Marxist in his personal politics but his historical accounts are quite "conservative", historiographically speaking.