You might want to spend some time browsing the exhibits at Cathedral of the Annunciation, Palace of Facets, or Pushkin Museum of Fine Arts. Grand Kremlin Palace, Terem Palace, and Cathedral of the Assumption are some of the local landmarks to explore. You can watch performances at Maly Theatre, Bolshoi Theatre, and Stanislavski and Nemirovich-Danchenko Moscow Academic Music Theatre if you're interested in local theater.