You can go into any upper division class and be expected to have already read the entirety of Paradise Lost and studied it in great depth. Since I essentially did the same thing, my recommendation is that you READ PARADISE LOST!!! You will thank me later, even though it is a difficult and lengthy read. For example, if you ever read Frankenstein, a huge component of understanding what's going on depends on a knowledge of Paradise Lost (this is the first book the creature reads - this is significant!).
Conrad's Heart of Darkness comes up pretty frequently, as does Twain's The Adventures of Huckleberry Finn. If you can read all the Shakespeare, go ahead, but at least read "A Midsummer Night's Dream," "The Tempest," and most importantly "Twelfth Night." The themes from these three Shakespearean plays are recycled over and over and over again (especially now in science fiction - those sci-fi authors love their Shakespeare) I know everyone says "Hamlet" and "Macbeth," but I have never needed to know them, so I say forget about them. The Great Gatsby by F. Scott Fitzgerald comes up nearly as often as Milton's Paradise Lost (which you really should read), so that will be required reading several times over. Beowulf is not necessary for your upper division work, but as an English major you should feel obligated to read the entire thing. Read the first book of Edmund Spenser's The Faerie Queene, but only the first book is necessary (just that one book is long).
Read a few short stories: Crane's "The Open Boat" and "To Build a Fire." Jackson's "The Lottery," Hemingways's "Hills Like White Elephants" and "The Snows of Kilimanjaro," Grab a dictionary of classical gods and goddesses, you don't need to read it, but you'll want it as reference. Familiarize yourself with the plots of Cervantes' Don Quixote, The Count of Monte Cristoe by Dumas, and Swift's Gulliver's Travels - they don't need to be read, but a basic understanding is helpful. The "General Prologue" "Wife of Bath's Tale and Prologue" from Canterbury Tales are sufficient to get by on until you have time to read all of the tales. You can read just Dante's Inferno, you don't have to read the other two parts just yet. If you know you Bible trivia as well, you're already ahead of most people in your classes. This is not strictly necessary if you don't know it though, because your professors will explain it for the benefit of everyone (they have to be politically correct and assume that not everyone reads the Bible).
You will definitely want to read the following if you have not done so already: Women's Lit: "The Yellow Wallpaper," "The Revolt of Mother," "The White Herron," "A New England Nun" and "Miss Spring Fragrance" (all short stories). The two major women's novels are Jame's Daisy Miller and Chopin's The Awakening, of which I find that later to be more useful. Just for some modern reading, Vonnegut's Slaughterhouse Five, which will probably be assigned anyway, and Kerouac's On the Road.
In terms of poetry, read the major staples (some are super short, other super time-consuming): Paradise Lost, The Lady of Shalott, T.S. Eliot's The Wasteland and The Love Song of J. Alfred Prufrock, Yeats' Second Coming, My Last Duchess, The Sick Rose (so short, yet so complex), Plath's Daddy, Lady Lazarus, and Colossus, Donne's Holy Sonnets and his other Sonnets, Dover Beach, Do Not Go Gentle into That Good Night, Mont Blanc, My Papa's Waltz.
Going just off the list of classic at, say, Barnes and Noble, half of them aren't worth the paper they are printed on, and are never required for helpful to have read. Hopefully this will be enough to keep you busy, I remember when I did the same thing before taking my upper division classes. This is pretty much all I can ever recall discussing right offhand in 7 years of English. If you only read one thing, make it Paradise Lost. Everything else will probably come up, but not the way that one will.
Good luck!