You might want to spend some time browsing the exhibits at Museum of History of Moscow, I.S. Turgenev Museum, or New Tretyakov Gallery - Museum of Modern Art. While you're exploring, swing by Pushkin House Museum on Arbat, Cathedral of Christ the Savior, and Arbat Street to learn a bit about the area. You can watch performances at Anton Chehov's Theater, The Man Theater, and Theatre on Malaya Bronnaya if you're interested in local theater.