Dang, wish I had seen this post back in May.
For MamaPata, what area of town are you living in? You of course have to do the Red Square/Kremlin. Go into any museum you find, even the small ones; you'd be surprised what you find in them. Try out the restaurants, lots of great food here.
Join the FB group Moscow Interacts for some links to good food and events. Ride the metro and stop at various stations and take a look around. Ride the new outer ring metro; it's still free I think till the 10th of October. It's above ground, so more to see than the "normal" metro.
For shopping, go out to Izmailovo/Измайлово. (Stop at Partizanskaya metro station and follow everyone else walking to Izmailovo. Do not stop at the metro station named Izmailovo.) Lots of good stuff there from the typical Russian gifts to actual antiques. Be careful of the foreigner discount, though.