Moscow has far more of a gay scene and homosexuality is far more accepted there than anywhere else in Russia. That said, it is certainly not comparable to a San Francisco, NY, London or Berlin!!
For many Russians, it is not an issue of accepting gay people, they are often completely oblivious of them! If they don't see open displays of it, they have no opinion on it. For many, gay is personified by camp. If you not a screaming drag queen, they will probably never suspect somebody of being gay! During the time I lived there, I never experienced or saw homophobia, not did I have any problems with Russian friends that I had made or close work colleagues. On the other hand, Russian gays tend to be far less open about their sexuality in public than you would see in big cities in the West. Of course, just to confuse matters, straight men are very touchy feely and it is not unsual to see soldiers walking down the street arm in arm with each other.
The scene is pretty discreet and compact in Moscow, but it is very much there and thriving, although you won't find anything resembling a gay quarter/village. Many hip bars and cafes will be tacitly gay friendly, most noticeably the ubiquitous
Propaganda/Пропаганда which has a gay night on Sundays. There are only ever a handful of gay night clubs, which may change owners, close down or turn straight! You will find that
Three Monkeys/Три обезьяны is usually pretty reliable. I have not lived there for a few years, so I do not know the most up to date haunts!
As you are staying there for a few months, you might find it useful to visit
Expat.ru and lurk in their forums, where you will find a wealth of information form foreigners living in Russia. There is also a useful web site
in English here which seems to be pretty up to date. I am sure you will have a great time in Moscow, the scene is usually pretty friendly and open and there is always quite a few gay expats knocking about (often appearing to make up about half the diplomatic community).
Feel free to PM me if you want to know more.