Just start here.
Penguin Classics. I've read just about all of them. If you're looking for "classics," they're on this list.
To give a quick few that I didn't notice on the above list I'll throw in:
Dante's Divine Comedies and La Vita Nuova
Herman Hesse's Steppenwolf, Narcissus and Goldman, and Siddhartha (especially the last)
Huxley's A Brave New World
Kafka's No Exit and other short stories (Metamorphosis, etc. etc.), also his The Trial
Camus' The Stranger
Bradbury's Fahrenheit 451
Beckett's Waiting for Godot
Bronte's Jane Eyre
Wilde's The Importance of Being Ernest
Fitzgerald's The Great Gatsby
... and I'm bored reading off my shelf. If you want more, let me know.