Personally I look on United.com first, starting a bit before the tickets will be available (330 days ahead). So looking just now, award booking IAD-MOW (Moscow all airports) for four travelers, I see quite a few options for August 4 outbound, routing on OS or LH via VIE or FRA or MUC. These are 1-stop flights in economy for 30k miles one way and about $45 in fees.
Given that that is Summertime, I would guess that September might have similar or better availability particularly if you book way in advance. You will not find a lower miles cost for those segments.
If you want to fly Business, I see a couple of options at 60k miles one way with either one or two stops (Transatlantic segment in Biz/Polaris, remainder in coach) and again around $40-60 in fees.
Personally I book as far ahead as possible to lock something in - it's easy and not too expensive to redeposit miles later if plans change.
I also use ExpertFlyer.com for specific things - for example if I book IAD-ZRH with a connection, I might use EF to check daily for saver availability on the nonstop. For Moscow you'll probably be connecting no matter what, so the UA site should serve your needs fine.