There is no end of excellent books for understanding and analyzing US society and politics. Here is a selection of those I have found the most useful.
For understanding US society and political system
Thomas Ferguson, Golden Rule.
Edward Bernays, Propaganda
Alex Carey, Taking the Risk Out of Democracy.
Noam Chomsky, American Power and the New Mandarins.
Morton Horowitz, The Transformation of American Law, 1870-1960.
Joseph Stiglitz, The Three Trillion Dollar War.
Jeremy Brecher, Strike!
Ralph Ketcham, ed., The Anti-Federalist Papers and the Constitutional Convention Debates.
Jacob Hacker and Paul Pierson, Winner-Take-All Politics.
Randolf Bourne, “War is the Health of the State” (essay)
Blackstone, ed., Cointelpro
The Pentagon Papers, Senator Gravel edition, including Vol. 5 (interpretation and analysis by editors).
Robert McChesney, Political Economy of the Mass Media.
William Domhoff, Who Rules America?
For understanding the US economic system
Dean Baker, Taking Economics Seriously.
Ha-Joon Chang, Bad Samaritans.
Joel Baker, The Corporation.
For grappling with other strains of Leftism, such as Leninism and Troskyism
Sheila Fitzpatrick, The Russian Revolution.
Paul Avrich, Kronstadt, 1921.
Maurice Brinton, For Worker’s Power.
Noam Chomsky, For Reasons of State.
For understanding US history
Howard Zinn, A People’s History of the United States.
Thomas Walker, Nicaragua.
Walter LaFeber, Inevitable Revolutions.
Gar Alperowitz, The Decision to Use the Atomic Bomb. (For a very succinct presentation of the same material, see Zinn, The Bomb.)
Marilyn Young, The Vietnam Wars.
Marilyn Young and Yuki Tanaka, ed., Bombing Civilians.
William Appleman Williams et al., ed., America in Vietnam: A Documentary History.
Bertrand Russell, Against the Crime of Silence: The Proceedings of the Russell Tribunal.
Ramsey Clark, War Crimes in Iraq.
Francis Jennings, The Invasion of America: Indians, Colonization, and the Cant of Conquest.
Noam Chomsky, Fateful Triangle: The United States, Israel, and the Palestinians.
Eric Foner, Reconstruction: 1864-1877.
Reed Brody, Contra Terror in Nicaragua: Report of a Fact-finding Mission, Sept. 1984-Jan. 1985.
Studs Terkel, “The Good War.”
Gary Nash, The Unknown American Revolution.
Paul Street and Anthony DiMaggion, Crashing the Tea Party.
Upton Sinclair, The Jungle.
James Loewen, Lies My Teacher Told Me.
Classics of dissident political philosophy
Rudolf Rocker, Anarcho-Syndicalism.
Daniel Guerin, Anarchism.
Daniel Guerin, ed., No Gods, No Masters
Emma Goldman, Anarchism
Noam Chomsky, On Anarchism.
Bertrand Russell, The Paths to Freedom.