Well the obvious attactions are redsquare the the kremlin. Definately gotta see those sites. Just be careful in the kremlin to stick to the designated areas and to stay off the roads, the police really don't like that. Also in red square it is possible to see Lenins body. You know when the opening times are because of the security checkpoint and the massive queue of people lining up to go through it. Make sure that you dont take photos, the police REALLY dont like that, also they like to shuffle people through in an orderly fashion so if you stop to look at his body for too long expect for a policeman to hurry you along.
For something a little different there is this club called the "hungry duck". This is a particuarly wild and debaucherous night club, expect to see a lot of nudity and sexually confronting russians. A fun night in my opinion!
One great experience though is simply to just explore the cities underground. Stalin made all the stations really really beautiful in order to create a "palace for the proletariat". An amazing site, just watch out for thieves.
I also swear by black cabs in Russia. Dont be afraid to stand at the side of the road and wave your arm at the traffic. When a car stops if you dont speak Russian point to a spot on your map and write down the amount you want to spend. By memory I think 200rubles was the going rate and that should get you to pretty much anywhere in town. Most cars will stop and take you to where you want to go. over 9 years ago