Tough question. First and foremost, you are there to meet a lovely lady. But what about filling in the gaps?
Well, the first time I went to Russia, my soon to be wife arranged to meet me in Moscow and we spent a couple days there before going to Ekaterinburg. While in Moscow, we stayed in a nice hotel close to the metro where we could get anywhere in city quite easily. The first thing she showed me was Red Square and the Kremlin. I think this is the one place all American men should go to see. Not only is it the place where we can associate our childhood and the cold war, but it is just a cool place to visit. The Kremlin has an amazing museum that you should see.
Other things to experience are riding the metro itself. If you’re curious, you can search out the Moscow McDonalds and I hear there is a Hard Rock Cafe there now. We also took a tour bus through Moscow and Lena interpreted for me. Saw the KGB headquarters and Moscow University. Was a nice ride anyway.
Another thing that was interesting was riding the train to Ekaterinburg. Russian trains are very nice. I paid for us to get a private cabin so we could get to know each other better during the 26 hour ride. It was well worth the extra money!
Wherever you travel to in Russia, there are many interesting things to do and see. As well as finding the many markings, statues and monuments of the communist era, you will usually find medieval history and lots of old cool stuff to see.
So, take advantage of your trip to Russia. Whether or not you find the woman of your dreams, there are lots of interesting things to see.