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.
Reading well-known works has two benefits: First, it gives you a common
frame of reference for things you may hear from lots of other people,
and second, it educates you in ways you may not have thought of. For
example: Most people have heard of "To Kill A Mockingbird." At a party,
one of your friends might express distaste for racial bigotry. While
none of your friends are bigots (hopefully), this friend is the one who
publicly took a stand against it. Saying, "You're the new Atticus
Finch" is a tremendous compliment to such a person, plus you will seem very literate saying it.