I have been putting together a list of the 10 best things to do in Moscow, that aren't in the standard tourist guides.
And I find myself starting with one thing that is ALL the tourist guides, the Metro. The best station must be Mayakovskaya - which has the most breath-taking light/flight/air themed mosaics on the ceiling. It is also the most 'lightweight' in design and looked after well; it has no adverts for example. The glorious part though, which is easy to miss, is the entrance hall to the north exit directly onto Tverskaya.
To find it when you get off the train, take the direction where the escalator goes down. And then rise up into a masterpiece of art and architecture.