What are the beliefs/main ideas of the Puritans, Colonial/Enlightenment, Romantic, Realist, and Modernist authors?

I'll answer the last part of this multi-pronged question—Modernism.
When we discuss Modernist literature, the names that are most often thrown out are those of Virginia Woolf, James Joyce, T. S. Eliot, Ezra Pound, and, on the less experimental but equally luminous end, Ernest Hemingway and F. Scott Fitzgerald.
Devastated by the carnage they witnessed during the First World War and disillusioned with a Western civilization that no longer melded with the values with which they had been raised, these authors eschewed those values. They reassessed notions of morality and included aspects of human life (e.g., sex, profane language, the body) that were previously unmentioned in literary texts.
Their choices with language and use of literary devices such as stream-of-consciousness narratives reflected the turn toward focusing on individual lives as they are actually lived, however ugly or unpleasant they might have been.
I would also argue that writers who are grouped into the Harlem Renaissance, particularly Zora Neale Hurston, Nella Larson, and Langston Hughes, were also Modernist writers. Their work also experimented with language (Hurston applied free-indirect discourse in her best-known novel, Their Eyes Were Watching God, a device in which a third-person narrator slips in and out of a character's consciousness to convey their experience) and sought new ways of depicting the conditions of people's lives—that is, without pretension or fear of offending God.