In the Kremlin, the cathedrals and armory (housing a vast collection of faberge eggs) are definitely worth a visit. get tix for the armory early as its busy.
on red square, lenin's mausoleum's gotta be done. st basil's cathedral is iconic from the outside, the inside was a bit unexpected. GUM, the old department store is also there, just a load of prada and lacoste shops tho.
As was mentioned, the metro is amazing, this link suggests a few stations: http://www.moscow-taxi.com/sightseeing/metro.html
the tretyakov gallery, if you're an ''''arty'''' person, is the best.
the cemetary behind the novodevichy convent is where many of russia's heroes are buried with phenomenal tombstones/gravemarkers
a drive around moscow at night to see a couple of the 7 sisters (7 stalinist skyscrapers e.g. MGU the university)
The Bolshoi theatre for some mandatory russian ballet?
some ideas at any rate.