I've been invited to visit in Moscow. Sounds like fun. This is
strictly private, with a new Russian Friend. Neither of us knows the
first thing about an American in Moscow. What do I need to know about
toilet paper, rough parts of town, getting a visa, and generally not
getting bummed out.

The first place you want to look is the US State Department
http://www.state.gov/travel

This site gives this info:
RUSSIA - *Passport and visa required. Tourist visa requires 1
application form, 3 photos, confirmation and voucher from travel
agency and/or hotel in Russia, cover letter from travel agency, and
processing fee. Business visa requires 1 application, 3 photos, cover
letter from your company, and letter of an official invitation from an
authorized Russian counterpart and visa processing fee. Visa
processing fee for business and tourist visas is $70 for 2 weeks, $80
for one week, and $110 for less than three working days.
Multiple-entry business visa (needs confirmation through the Russian
MFA) - $200 to $450 processing fee. Fees must be paid by money order
or cashier's check only. Transit visas required when traveling through
Russia, even if you remain on the plane. Provide SASE for return of
passport by mail. HIV test certificate required for anyone staying
over 3 months. For additional information, contact the Consular
Section of the Embassy of Russia, 2641 Tunlaw Road, N.W., Washington,
D.C. 20007 (202/939-8907 or 8918) or the nearest Consulate General: NY
(212/348-0926, 0955, 0626), CA (415/928-6878) or WA (206/728-1910).
Internet: http://www.russianembassy.org

There is also a Travel Tips for Russia.

http://travel.state.gov/tips_russia.html

You will also want to go to the Russian Embassy web site
www.russianembassy.org

This site will have info like:

A tourist visa can be issued for not longer than one month!

    When entering Russia with a tourist visa, you are required also to
present at the border check point your tourist confirmation, voucher
from your hotel and a return ticket with a fixed date of departure
from Russia.  You also will have to prove that you have sufficient
funds for your travel.

I also found this site that has some good info. It is a Russian Travel
Agency.

http://www.sv-agency.udm.ru/svfiles/living.htm

When ever you are out of the US you should check on medical care
avaiable

http://www.mdtravelhealth.com/destinations/europe/russia_major_cities.html

Have a nice trip and have fun.

I might add using a expediting service.  Here is a list off Yahoo:

http://dir.yahoo.com/Business_and_Economy/Shopping_and_Services/Travel_and_Transportation/Passport_and_Visa_Services/

I used passportsandvisas.com (Google advertiser)