So you are interested in studying music? You are absolutely right to look here!
Belarussian State Academy of Music (formerly k.a. Belarussian State Conservatory) is the best place in the former Soviet Union to apply. Here are the reasons:
First, because it is comparitively cheap. The tuition fee is $2000 per academic year. If you don't speak Russian you can opt to take an intensive course of Russian and classes on your instrument for the first year (pre-year), which will cost you only $1500. The tuition includes dormitory, a 50% discount on public transportation, and free cab fare (joking about cab fare :)))).
The second reason is that teachers here are as good as those in Moscow (a lot of them graduated from there) and are considered as highly qualified and experienced.
Third reason, and this can be related to any of the Universities enlisted here, is that the cost of living here is much lower than in Moscow or Saint-Petersburg or any Russian city. You can easily live here for $500/mo. including everything from your own apartment to dining at restaurants.
So, the address of the Academy is Belarus, Minsk, Internationalnaya str., 30 phone number is (+375-17) 227 11 03. This is the number of the foreign student department. To talk about your studies, you should ask for Lubov Alexeivna. She is a nice woman who speaks a little English. It's her job to deal with foreign students, and she will be able to give you more specific information on instrument majors and degree information.