There are a great number of classics that you'll never find recently published in hardcover so, for that and financial reasons, I wouldn't recommend focusing on them (you can certainly find a lot of essentials in the Everyman's Library if you're still inclined that direction.) Both Penguin Classics and Oxford World's Classics are consistent go-tos for the breadth of their catalogues and translations in good standing -- the current imprints look quite handsome on the shelf as well. NYRB Classics is probably the independent publisher of record at the moment with lots of exciting modern classics and newly translated titles coming out every year.
If you're just starting a collection, I recommend keeping an eye on the variety of periods and cultures your books are coming from: most readers rarely escape the orbit of a few familiar names, times, and places.
An example of a core collection I would recommend:
Odyssey by Homer Metamorphoses by Ovid The Thousand and One Nights Don Quixote by Cervantes Gulliver’s Travels by Swift The Sorrows of Young Werther by Goethe Pride and Prejudice by Austen Frankenstein by Shelly The Red and the Black by Stendhal Eugene Onegin by Pushkin Le Père Goriot by Balzac Collected Tales by Gogol Collected Tales by Poe The Count of Monte-Cristo by Dumas Vanity Fair by Thackeray Wuthering Heights by Emily Brontë Jane Eyre by Charlotte Brontë The Scarlet Letter by Hawthorne Moby-Dick by Melville Madame Bovary by Flaubert Great Expectations by Dickens Fathers and Sons by Turgenev Les Misérables by Hugo Notes from the Underground by Dostoevsky Alice’s Adventures in Wonderland by Carroll Middlemarch by Eliot Anna Karenina by Tolstoy The Portrait of a Lady by James Collected Stories by Maupassant The Way We Live Now by Trollope The Adventures of Huckleberry Finn by Twain Germinal by Zola Hunger by Hamsun The Picture of Dorian Gray by Wilde Tess of the D’Urbervilles by Hardy Collected Stories by Chekhov Dom Casmurro by Assis The Immoralist by Gide Nostromo by Conrad The House of Mirth by Wharton Sons and Lovers by Lawrence Kokoro by Soseki Collected Stories by Akutagawa A Portrait of the Artist as a Young Man by Joyce Main Street by Lewis Red Cavalry by Babel Zeno’s Conscience by Svevo The Magic Mountain by Mann The Trial by Kafka The Great Gatsby by Fitzgerald In Search of Lost Time by Proust To The Lighthouse by Woolf Steppenwolf by Hesse The Radetzky March by Roth Kristin Lavransdatter by Undset As I Lay Dying by Faulkner Independent People by Laxness
which leads you up to the Second World War where matters of taste and preference begin to enter more prominently into the argument. For theater you'd want core dramatists: Aeschylus, Sophocles, Euripides, Marlowe, Shakespeare, Molière, Racine, Ibsen, Strindberg, Wilde, and Shaw. Poetry suggestions I'll leave to more dedicated fans.