Cosmonautics Museum: Super cool collection dedicated to Russian/Soviet space exploration, from Sputnik to the Space Station.
If you want a truly special experience, think about visiting Moscow in mid-January, just after the Russian Orthodox Christmas. The city is most likely sporting a beautiful blanket of snow as it twinkles with thousands of Christmas lights. Some people feel that visiting Moscow at this time is the best way to experience it: Dr. Zhivago-style. With appropriate winter gear, you won't even notice the weather.
For an action-packed night on the indoor ice, I suggest taking in a local professional KHL (Kontinental Hockey League) hockey game at the VTB Arena (for the Moscow Dynamo team) or CSKA Arena (for CSKA, the "Red Army" team). Both arenas are easily accessible by public transport.
LavkaLavka: Put away your preconceptions about Russian food because this tiny restaurant is one of the shining stars of Moscow's farm-to-table cuisine movement. You'll know who grew the vegetables on your plate or whose cow gave the cream or whose chicken laid the eggs for the meringue on your dessert. There's also a small retail shop in the courtyard carrying those locally-produced items. Why not pick up a jar of the delicious "forest trees" honey to take home in your suitcase? (I did!)