If you can go to Red Square at night, do. If you can't, go during the day. Either way, definitely go see the quintessential center of Moscow. While foreigners go there (and get their photos taken with the Lenin and Stalin impersonators by the archways on the side opposite St. Basil's), Muscovites also frequent Red Square but usually after sundown. The mall next door, GUM, and some nearby streets are lit up, and it is simply gorgeous to be there once the sun goes down.