Well, you should definitely go to Moscow and St. Petersburg if it's your first trip. In Moscow there are several places around Red Square/Kremlin that are must-sees like St. Basil's Cathedral and the Bolshoi Theatre. If you are interested in Russian art, go see the Tretyakov Gallery which is the largest collection of Russian art. I also recommend the view from Moscow State University, it is the best view of Moscow at night. While in Moscow, be sure to experience the subway stations, they are each unique works themselves with elaborate construction.
In St. Petersburg, you should see the Hermitage which is one of the greatest collections of art in the world. Also, go see Catherine's Palace while you are there. If you have the oppurtunity, you can take a night train from Moscow to St. Petersburg and back. That way you don't waste a day traveling and it's quite a fun experience too, just have a travel agent or your hotel book a private cabin for you.
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