You'll want to browse the collections at Moscow Museum of Modern Art, State Historical Museum, and Andrei Rublev Museum of Early Russian Art. Palace of the Romanov Boyars, Bolshoi Theatre, and Resurrection Gate are well-known local landmarks worth a visit. You can watch performances at Maly Theatre, Stanislavski and Nemirovich-Danchenko Moscow Academic Music Theatre, and Moscow Arts Theatre if you're interested in local theater.