Moscow

I'm not sure about a travel agency, as when I was there three years ago I did it all myself, but was staying in the city for four days. We flew Aeroflot from Rome to Moscow and back. They were a fine airline, comparable with any other carrier, nothing special though. Which airport are you flying to? I will guess Sheremetyevo, since that is where we flew to on Aeroflot. Aeroexpress trains run from the airport to Belorussky Railway Terminal from where you can take the metro to Red Square. The square is free, so no money needed for that. St. Basils was not expensive, maybe a few dollars equivalent in Rubles. You might be able to see Lenin's tomb, but there is often a long line, it is free. Lunch should be fairly reasonable, but with your time frame, plan on eating in the tourist area, so budget more than you think. Get cash at the airport, few spots accept credit cards and I would not try to find a bank in the city with your limited time. With only 10 hours, your time is very limited. It took me two hours just to get through customs. So plan on: 2 hours customs 1 hour to get luggage and stow (or bring with you- not recommended) 1.5 hour to get to Red Square 2.5 Hour to tour around Red Square and eat 1 .5 hour to get back to airport
Balance of time to make sure you do not miss your flight and get back through security, which WILL be a hassle. PM me if you would like more information.

I know one travel company which arranged our tours in Moscow and St. Petersburg, they had also sent us visa support for the trip. Their website address was palladium.spb.ru as far as I can remember.