Domodedovo Airport is the flagship of Moscow's new modern airports. It is located in the south east and is accessible by public transport buses, marshrutkas (fixed route mini-vans) and the Aeroexpress train. While not the size of Heathrow by any means, this is a modern airport; they even have xray scanners to see if you are concealing weaponry or knitting needles on your person.
Tickets for the Aeroexpress train can be bought at the cash desk or via the ticket machines, near the exit to the trains. From Paveletsky railway station, follow the signs to metro Paveletskaya (brown Circle line) and it is only 10 minutes by metro to reach the city centre.