I found in my travels that climate at different places is never as bad as people make it out to be. Basically, the weather during your trip is a matter of luck. Russia has a continental climate, which means that it is cold in the winter and hot in the summer. Consequently, the best times to visit are late spring, early summer and early fall. Russian winter may have some very beautiful days that are sunny and frosty or snowy and romantic. Winter is also a peak of the theatre season.
However, the worst thing about Moscow from November to April is street cleaning. Bring waterproof footwear for there is no way to avoid a muddy slush covering the streets. They also use salt very liberally. I usually carry a portable shoe cleaning brush or rag in my pocket to clean the white residue off my boots when coming form the street. Late summer is vacation time for most Russians. Expect the city to be empty and there may be a certain lack of entertainment in July and August when most of the locals leave Moscow.
To learn more about Russian weather go to:
Russia's Weather - http://meteo.infospace.ru
Meteo data of Russia - http://www.meteo.ru
Today weather: