As a foreigner, most of us are usually afraid to do new things which we don`t know, such as buying a car. We are mostly afraid of the diplomatical and procedural things. Instead, it is not such a hard thing and i can say that it is even easy after buyig it.
Firstly you have to decide which car to buy, as well as deciding buying a brand new car or used one. I would offer to buy a new car beacuse there is not much price difference. You should buy from the offical sellers in the city. The others sellers are mostly more expensive and they dont give a trustfull image to you. You have to give an order 3-4 months before to get the car with your specifications. or you can buy the car waiting in the saloon which will not fully satisfy or be over your expectations.
For foreigners there is a different place for car registrations. It is located near Tsaritsno Metro. To do prodecural thing you can find somebody(some Turks working in Moscow long time as driver for example) who would help you in such things with some money. Also in this process, payoff to some people is needed to get your things fastly done.
While driving car you have to carry 4 documents with you.
1) The translation of your driving license in Russian and the original driving license.
2) The OSAGO document, which is the govermental insurance of the car.
3) The registration document of the car to your name.
4) Ownership document of the car.
Also you may carry the private insurance document with you.
Buying car and registering is the only two steps firstly to buy a car. If you have money and will to have a car, i would offer you to buy a car and not to wait for it. Your life really changes in moscow after having a car...