Having been to Russia several times (and not speaking more than a few words of Russian) here's my advice:
-Lose the baseball caps
-comfortable, earth-tone clothing (nothing screams "American" like bright colors)
-learn to watch your surroundings without stareing, even in the US this announces you're a tourist.
-we tend to be louder, smile/laugh loudly more and need more "personal space" than the locals
-try not to have that big "tourist" camera/s hanging around your neck
-I'll second lonelyplanet as a good resource, check it out
I apologize if you're a seasoned traveller and already know this, but remember that you will also represent our country in an area of the world that already views us poorly so: be friendly without being condescending, and understand that this is a (very) different culture with different attitudes about customer service, public conduct etc...getting uppity because "that's not how things are done back home" or being interested in "your cute little culture" is insulting to the locals, can get you in trouble, and feeds in to general attitudes about what an "American" is. Also, I know what the "warning signs" are when I'm walking through SoHo at 2AM, but I found that I couldn't pick up on the warning signs of a small village outside Moscow...stick with your local friends/guides, follow their lead, avoid talking if you get around the drunk kids by the beer kiosks late at night and try to stay relatively sober if you're going to be out on foot at night (reasonably difficult there, but... ) Bottom line, you won't be able to prevent being pegged as a "foreigner" and your shoes/haircut/clothing cut will probably peg you as American before you even open your mouth. Try not to add "obnoxious" to that, and you'll do OK.
For self-defense: if you don't know how to throw hands already, it's probably too late to learn for this trip. Keep your head about you, watch for overt problems--being followed by 1+ guys, harrassing tones from young drunks, someone "sizing you up"--stay in groups, and AVOID PROBLEMS. The umbrella or magazine idea is better than nothing, but they'll be fairly ineffective if you don't already know how to use them as a weapon. If you don't know how to fight, then situational awareness will be crucial for you--just remember that an over-alert posture can also telegraph fear/insecurity/readiness for victimhood.
Customs doesn't care what you wear, so: loose, cool , comfortable clothing--you're going to be on a plane for upwards of 8 hours. Shoes that slip on/off easially (good for security in the US and for removal in the plane). A shirt with a breast pocket big enough to carry your passport and boarding pass; you typically have to show these as several points to several people throughout your trip through the airport...having it readily accessible and not having to fumble through your bag for them makes the trip a little easier.
And FWIW, I tended to get pegged as German more than I did as American...go figure.
In my experience, most of the people will be friendly and the exposure to such a different environment (can) be great. Have a fun trip!