Say I am an author around the 1850's in the USA, and I just wrote a book, and I want it published. How would I go about that? Were there any distinguished publishing companies? Do I just make a lot of copies and distribute them in book stores?

There was a fairly flourishing US publishing industry by the 1850s. For example, Melville published with Harper and Brothers (which is is extant as HarperCollins), Poe with Putnam (also still extant as an imprint of Penguin), Hawthorne and Thoreau with Ticknor and Fields, Emerson with Philips, Samson.

Source: William Charvat, Literary Publishing in America, 1790-1850. U of Massachusetts, 1993