After you clear arrivals area, go upstairs and exit to your right. Follow the signs to the AeroExpress train. It's a 10 minute walk to the platform covered 90% of the way. Buy a ticket for the next train for 250 Rubles and wait till the train boards. I suggest the Starbucks as a great place to wait.

Take the train till it stops and you are at the Savyolovksaya Metro station (Grey line, 1 stop and a walk through the station from the Brown Ring line).

Note the Metro does not go to SVO. This train opened late last year and makes the trip to the city much, much, much easier than anything previously offered.

Good luck!