The easiest and fastest way to get to-from Moscow and airports is to use express train Aeroexpress. Here I will give detailed information on how to locate the stations of Aeroexpress around city and how to buy tickets to this train.
For all my guests especially those who are going to visit Moscow for a private tour during their layover here I always advise to use only Aeroexpress trains. The main purpose for it to avoid any traffic from-to airports and in 35-45 minutes you can get to the city center from the airport. I have to note that all the airports of Moscow are not in the city but away from it. Usually it can take you around 40-60 minutes by car to get there. But unpredictable traffic in Moscow can make your way to the airport from 30 minutes drive to 2-3 hours waiting in the traffic jams.