Novodevichy Convent is one of the most unique experiences we had in Moscow. Some of the buildings date back to the 1500s. They sell tickets to foriegners for a couple of dollars (Russians are free). It is located in a loop of the Moskva. The architecture is fascinating, but the best thing about the visit here is the absolute calm and peace.
Even though we are not Orthodox, we enjoyed seeing the monastery, complete with monks and nuns. Being Protestant, we may have enjoyed it more because it was a completely unfamiliar and interesting sight. Don't come with an intent to hurry your way through--enjoy the beauty and tranquility.
Be sure to see the Smolensk Cathedral. The interior has beautiful frescoes that date back the the 1500s. There is a five-tiered iconostasis from the late 1600s.
Also see the Cathederal of the Dormition. At the time we visited, daily services were held.
Above the main enterance is the Gate-Church of the Transfiguration. Next to it, you can see Lopukhn Palace.
There are two cemeteries. Of particular interest is the new cemetery. Some of the graves include Nikita Khrushchev, Nadeshda Allilujeva (Stalin's first wife), and Anton Chekhov. Also buried there are a number of writers, musicians, and politicians.