Learn the alphabet before coming to Russia. Names of streets, metro stations are usually written only in Russian. Several basic words and the alphabet will help you to get around Moscow on your own.
Use the metro. Moscow is a big spread out city and many of the sights are not in walking distance from each other. The traffic in Moscow is slow and unpredictable so metro will save your time. Moscow metro is cheap, clean, fast and some of the stations are breathtaking. Do take pictures in Moscow Metro.
Beware of pickpockets in crowded public places. Watch out for your cameras, money and documents.
Keep an official copy of your passport. The police are allowed to approach to anyone and check the documents. The usual advice is to carry your passport and visa always while in Russia. If you're afraid to lose your papers - use photocopies.
Negotiate the price before taking a taxi. Even if it is an official taxi and the driver says he has a meter. There have been cases of meter showing astronomic prices after 5 minute driving.
Bring the necessary adaptor for electronic devises if needed (Russia uses 220v)
Take off your hat ( if you're a man) when you enter a church. Religious women usually put on scarves or hats before entering a church but this rule is not a must. Put on modest clothes if you're planning to visit a church or a monastery (no open shoulders or miniskirts or shorts)
Give an odd number of flowers to a person. An even number of flowers is given only to a dead person at funerals
Explore numerous Moscow museums and theaters
Experience Moscow night life. Moscow is a 24-hour city where life never stops. Many restaurants, bars and clubs work all night. This is a chance to meet nice and friendly people of Moscow!
Don't drink tap water. It's safe for teeth brushing but not good for drinking. It's better to buy some bottled water.
Don't bring old used dollars or euros. Russian banks might refuse to exchange them. It's better to bring new EURO or USD notes.
Don't bring travelers checks. It will be a pain to find a place to cash them and it'll take you a lot of time.
Don't use ATM'son the street or in the metro. There have been scandals with cards and pin numbers being stolen with ATM's at these locations. It's safer to use ATM's at hotel lobbies or inside banks.
Don't take pictures inside churches.
Don't place your empty bottle on the table. It's bad luck. Empty bottles should be placed on the floor
Don't sit at a corner of the table if you're single. Tradition says you might never get married.
Don't ask 'how're you?' if you don't have time to listen to the reply (which might be long). In Russia this is a real question but not a greeting.
Don't be afraid of Russian winter. Wrap up warm and enjoy the snow and empty halls of Moscow museums.