I enjoy reading classics not just because there is a good reason they are "classics" (most of them are really, really good reads) and are part our common cultural heritage (enables you to understand references and participate in cultured discussions), but also because I get transported to different times, places, cultures. If the latter appeals to you, then go for a variety of time periods and authors. Don't just pick some titles off an English-focused list.
Definitely read some Dickens (Tale of Two Cities, David Copperfield, Great Expectations are my favorites) and some of English "lady authors" (Pride and Prejudice, Jane Eyre, Wuthering Heights). Less popular, but still one of my favorites is W. Somerset Maugham (Of Human Bondage, The Razor's Edge). Also great, Trollope's Vanity Fair. Also, try at least some Shakespeare.
Some French classics: Balzac (Pere Goriot, Lost Illusions), Zola (I love Germinal), Hugo (my favorite is Les Miserables), Flaubert (Madame Bovary). I'm not a huge fan of Dumas, but I like Queen Margot. If you like adventures, some of his other novels, Three Musketeer, Count of Montecristo, may appeal to you.
Read some of the Russian classics: Dostoevsky (Crime and Punishment is my fave), Tolstoy (I prefer Anna Karenina over War and Peace). A more recent Russian classic, Doctor Zhivago by Pasternak. Lots of people don't like it, but I loved it.
Try some Latin American literature. Garcia Marquez (100 Years of Solitude or Love in the Time of Cholera), Vargas Llosa (Aunt Julia and the Scriptwriter, The Feast of the Goat), the short stories of Jorge Luis Borges and Julio Cortazar. If you like biographies/autobiographies, read Neruda's I Confess that I Have Lived.
Try to get through some of Don Quixote, as it is probably the world's first great novel, or as some academics call it, the first "modern novel." It is huge, but it's written as a series of episodes or adventures, so you don't have to get through all of it to get a taste. It is actually great fun.
Some American classics that I like: The Last of the Mohicans, To Kill a Mockingbird, For Whom the Bell Tolls (or The Sun Also Rises), The Jungle, Sister Carrie, The Grapes of Wrath, Catch 22
Well, that's quite a list, I got a bit carried away ...