I would recommend you to just get a taxi it would be easier and safer. Luggage screams rich tourist and you don't need that. there are lots of stairs and poor people begging on them asking for hand outs. if you want to see the poor side of Moscow, then this is the best way to see it.

Take Aeroexpress shuttle train and you would come to Paveletskiy railway station in 45 minutes. Then, change to "Paveletskaya" subway station of Circle or Zamoskvoretskaya line. The price for the Aeroexpress if 320 Rub.

If you are one or two persons without to much luggage, the Aeroexpress train is the best option from any of Moscow airports. It runs always on time a brings you to one on the trains stations on the Moscow metro circle line. If you are more people I would recomend to use a taxi but check the price before you enter. It should be around 1000 max to the center. The problem can be the trafic which can prelong your taxi trip.

Anna is correct. I would really thin twice befor takeing your luggage. The Metro moves thousands of people and there are also many stairs to deal with. just got back from moscow last week

The easiest way I know is taking the Aeroexpress. It costs 320 rubles and stops at the metro. It goes every half an hour and takes 40 minutes to arrive at the Moscow train station. You'll arrive at Paviletskaya Station (a train station). The metro quite near. Well, it's not so easy to go to the metro with luggage, but it's possible (I did it). You also have the option to store the luggage at the train station you arrived (it costs 83 rubles per day).