Ah.. it used to be much easier to answer this question, even some twenty years ago. There used to be something of a recognizable canon of great books that any person who wanted to consider themselves generally cultivated was supposed to know. We could start with British literature, where one had to know Shakespeare and Miltons works, then the main Romantic poets (Wordsworth, Coleridge, Keats, Shelly, Byron) then some Dickens of course, and perhaps for the modernists Joyce and Virginia Wolf. In French literature you needed to know som Racine an dMoliere, the great 19th century novelists, Balzac, Stendhal Flaubert, and the main poets, Baudelaire, Mallarme, Rimbaud, and then Proust.  You needed to know Dostoevsky and Tolstoy, Perhaps some Pushkin, and Melvilee Poe, Emerson and Dickinson, Faulkner and Hemingway. if you wanted to go beyond literature you were expected to know something about Plato and Aristotle, Some Kant or Hegel, some Darwin and Newton, Freud, Marx and Nietzsche. However, these days are no more. The very idea of an established accepted canon deciding on what is more important and what is less, what one needed to know and what one could more or less ignore, has been increasingly put in question, as always being the creation of this or that interested group, with its own motivations, trying to spread its values and dominate through them, repressing the created works of other value systems, or even repressing the creative legitimacy of other groups, resulting in the lack of works that will speak for these groups, etc. The last two decades thus stand under the sign of the attempts to retrieve some forgotten and repressed works, and to revise or even completely destroy the very idea of the canon, and thus the very idea of what an education should be. it is not clear what meaning being read well today still have. Personally I think that you could do no wrong by following the "old" canon for your education, nevertheless being aware that it is no longer, and that there is no longer, a firm concept of being well read, and you should try to expand your reading beyond the old canon - which still serves a solid base serious thinking, by following your interests in the directions they take you.

Excellent answer Dave.
Dave's post gives you a lot of ideas what to read. Of course he couldn't mention everybody, and the "former" canon is quite vast. I would recommend you go to your local bookstore (or Amazon/Barnes&Nobles etc.) and take a look at any book from the Norton Critical Edition, or the Penguin classics. All of them are classics, and you can take a look, read the back-cover and bit of the book (sometimes in Amazon you can search inside book and read a bit like that) and see whatever strikes your fancy. For instance, Dave didn't mention the Bronte sisters or Jane Austin. Perhaps they are bit less at the center of the canon, but you might really like reading them, and that is the point. You want continue with your quest unless you really enjoy it, so I would recommend trying to choose books you think you'll like.
Also, try to see how do you connect to books. Norton's Critical Edition has some commentaries on the book/period/author which you might find interesting. Perhaps not at all. There are critical books which deal with authors, for instance Nabokov on Gogol, which give you a track into analysis of books and thus could make it more interesting to read. But then again, you might not like it at all and prefer not to think too much about the books.

I would recommend , if you want to really be "well read" to also read critical essays on books as I think it helps to open you to an understanding and options of what authors try to do. Of course, also here I would recommend only reading the good people as otherwise it is conteminating and leads you in the wrong direction. For advice on people to read here, I would send it back to Dave who seems to know more about this than I do, so he can probably direct you to some good people.

To mention some more names (who Dave forgot to metion) of great authors you might want to consider: Kafka, Thomas Mann, Musil, Kleist, Gogol, Voltaire, Montaigne and well, many many more.

Last point. As I mentioned, to keep reading you would need to connect with it and like it, and each person has different ways on how to do that. You might decide you prefer a general view and read a ton, and you might decide maybe you want to know a particular person and read all their books and some books on them. (probably some of both possibilities is good). You probably already have an idea how you connect to topics and what brings you to continue with things.

Enjoy.