There are two main international airports in Moscow: Domodedovo Airport and Sheremetyevo Airport. Mostly all the tourists come to Moscow using these airports. And the main question for them how could they get to Moscow center or their hotels.
There are several ways how you could get to Moscow center.
The easy one for you will be take a taxi from the airport (use the official taxi services, you can find their counters in the airport). But this way could be a little bit expensive for you so you should be ready to pay from 80USD to 150USD to get to your hotel in Moscow. Also you can ask your hotel manager if they could organize the transfer service to/from airport. Regarding to traffic and time you will get your hotel in 1 to 3 hours.
NB: We usually use this service to have a taxi to/from airport: www.taxi-sms.ru/ (they have reasonable prices for us). But we are not sure if they could speak English. If you need transfer to Moscow we could organize it for you!
The cheapest way to get to Moscow will be to use Aeroexpress. This company provides the rail transportation services between Moscow rail terminals and Moscow airports. It is easy to find their counter in the airports and you can also buy their tickets beforhand using their website (engl). The best part about Aeroexpress that you could buy not only tickets to Aerotrain (train between airport and rilway station) but also metro tickets (Aeroexpress + Metro) or service like transfer from airport to rail station by Aerotrain and then from the rail station to your hotel by taxi (Aeroexpress + Taxi). Transfer time from airport to railway station around 40-50 min.
You could find the rates for different tickets on their website.