I have been taking a class on Russian cinema. Here are some great movies you should watch.
Battleship Potemkin-Sergei Eisenstein's most famous work, and, incidentally, Hitler's favorite movie. The tale of a mutiny on a battleship, this film incorporated Eisenstein's revolutionary montage techniques.
Man With a Movie Camera-Dziga Vertov's demonstration of his Kino-eye idea. This is probably the best movie ever made. Images of Russia during the NEP.
White Sun of the Desert-A tale of the Russian Revolution in the East. One of the most popular movies ever in Russia, and watched as a ritual by cosmonauts before they go into space.
Hedgehog in the Fog-A short film by Yuri Norstein, one of Russia's most celebrated animators. Youtube here, but try to watch it on a big screen in high quality.
http://www.youtube.com/watch?v=dRsXU4Q6a0Q Solaris-Probably Andrei Tarkovsky's most celebrated film. A science fiction story set on the spacestation orbiting the planet Solaris. Slow, mysterious, troubling, but ultimately rewarding.
Stalker-Another of Tarkovsky's work, and probably my favorite movie. Three men go into a mysterious Zone seeking a room that grants wishes. Based loosely on the novel Roadside Picnic by the Strugatsky brothers.
Some assorted films that are good, but not worth giving a synopsis for. Earth, Chapayev, Ivan the Terrible Parts one and two.