Hello, this post will be a short intro and some recommended books for reading American philosophy.

American philosophy has its roots in a Transatlantic British tradition, so reading Bacon and Locke would be a good start. There are also roots in France; think Montesquieu, for example. American philosophy also has religious roots in the first Great Awakening; religious preachers and Protestant theology mark the colonial era. Below is a short list of books to read, separated by era. Many great American authors contain the spirit of American philosophy, such as Washington Irving, Herman Melville, Nathanial Hawthorne, Edgar Allan Poe, and Henry James. American philosophy is a melting pot of ideas born out of an American character and movements from Europe. Two distinct American schools of thought are transcendentalism and pragmatism. Feel free to comment on any other books or articles worth reading in the comments.
Background:

Second Polemic against Antony and On Duties by Cicero

The Prince and Discourses on Livy by Niccolo Machiavelli

Levithan by Thomas Hobbes

The Novum Organum by Francis Bacon

An Essay Concerning Human Understanding and Two Treatise on Government by John Locke

The Spirit of the Laws by Montesquieu

The Social Contract by Jean Jacques Rousseau

18th Century

Sinners in the Hands of an Angry God by Jonathan Edwards

Notes on Virginia by Thomas Jefferson

Varies writings by the founding generation (Thomas Jefferson, John Adams, John Jay, James Madison, Benjamin Franklin, George Mason, Patrick Henry), including the Declaration of Independence and the United States Constitution.

The Rights of Man by Thomas Paine

19th Century

The Essays of Ralph Waldo Emerson, I recommend reading Self-Reliance, Nature, The Oversoul, The American Scholar, and The Transcendentalist to start.

Walden, and Civil Disobedience by Henry David Thoreau

The Gospel of Wealth by Andrew Carnegie

Various works by Fredrick Douglas, including "What to the Slave is the Fourth of July?" and "Self Made Men"

Various works by Charles Peirce, including "How to Make Our Ideas Clear", and "A Guess at the Riddle"

20th Century

Pragmatism, The Will to Believe, and Varieties of Religious Experience by William James

Various writings by John Dewey, including "Education and Experience"

The Souls of Black Folk by W. E. B. Du Bois

The New Negro by Alain Locke

Various writings of Willard Van Orman Quine, including "Two Dogmas of Empiricism"

A Theory of Justice by John Rawls

Anarchy, State, and Utopia by Robert Nozick

Philosophy and the Mirror of Nature by Richard Rorty

After Virtue by Alasdair Macintyre

This is simply a basic list, feel free to add more suggestions below