Moscow -
I recommend spending a day (or as much as you can handle) just riding the Metro. Go on the circle line, get out at each station and have a look at them all. Each one is different from the next, but just as stunning. And you can stay on the Metro for as long as you like and only pay once!

I also recommend a visit to VDNKh (Metro Stn, erm, VDNKh), now known as VVTs (All Russia Exhibition Centre). Once a showpiece for the USSR's achievements in fields such as industry, agriculture, transport, aviation, etc, the place is now little more than a market, but it's still an interesting place to browse around. The original pavilions, statues and fountains are still there, and it's worth a visit for these alone - the architecture is amazing.

Also try Izmailovskii Park on a Saturday. Can't remember if it's Metro Stn Izmailovskii Park or Izmailovo, but it's the one with statues of partisans on the platform - you'll know it when you get there. Then just follow the crowds through the hotel complex. It's a bit tourist-y, but still a fascinating browse, with lots of Soviet memorabilia. Also there is a more "real" non-touristy book market (i.e. Russians go there, not only tourists).

If you're into museums (particularly WW2 history), Poklonnaya Gora (Metro Stn Kutuzovskay) is the place, one of the best museums I've ever seen. Great static displays in a nice park, too.

Enjoy your trip!