The vast majority of foreign travelers arrive in Moscow by plane at two major international airports:

Sheremetyevo Airport-Terminal 2 (SVO)
Operating airlines 
Online shopping in Moscow Duty Free 
Reserve a taxi at reasonable rates
*There's no metro stop or train station at this airport.

Domodedevo International Airport (DME)
Check flights status
Check international & domestic flight schedules 
Operating airlines 
Train to city center (Paveletskaya Metro Station) 
Official taxi carriers for Domodedovo airport 

Most flights to desinations within Russia and CIS countries are served by Moscow's other airports:
Vnukovo Airport (VKO) 
Sheremetyevo Terminal 1 (SVO) 
Bykovo Airport (BKA)

Also check the websites of major Russian airlines:
Aeroflot - Russian International Airlines 
Transaero 
S7 Airlines (former Sibir) 

ARRIVING BY TRAIN

Check trains and routes within Russia and CIS
*When doing a city search in English, the city names must be transliterated properly. For example, instead of Moscow type "Moskva", instead of Saint Petersburg type "Sankt-Peterburg". Alternatively, try to search by the initial letters only. Note that some destinations like Nizhny Novgorod and Ekaterinburg still exist in the Russian Railways system under their old names: "Gorkii" and "Sverdlovsk"   

Moscow has eight major train stations. All are served by the Moscow Metro and located near the center of the city. (See map of central Moscow.) Taxis are readily available outside most stations but beware they may be expensive. If you're on an international train, you'll go through customs when you cross the border, not at the station.

Belorussky Station: Located 500 m away from the Sheraton Palace Hotel; serves Kalliningrad, Lithuania, Belarus, Poland, Germany, the Czech Republic and some trains to Latvia.
Address : 7 Tverskaya Zastava Ploshchad
Phone : 251-6093, 973-8191
Metro: Belorusskaya

Kazansky Station: serves Central Asia, Ryzan, Ufa, Samara and Novorossiisk.
Address: 2 Komsomolskaya Ploshchad (See Map)
Phone: 264-6556
Metro: Komsomolskaya

Kievsky Station: serves Western Ukraine and Southeastern Europe.
Address: Ploshchad Kievskogo Vokzala
Phone: 240-1115/0415
Metro: Kievskaya

Kursky Station: serves Southern Russia, Caucasus nations, Eastern Ukraine, and Crimea.
Address: 29 Ul. Zemlyanoi Val (See Map)
Phone: 916-2003, 917-3152

Oktyabrsky (Leningradsky) Station: serves Estonia, Finland, St. Petersburg and northwestern Russia.
Address: 3 Komsomolskaya Ploshchad (See Map)
Phone: 262-9143
Metro: Komsomolskaya

Paveletsky Station: serves Voronezh, Tambov, Volgograd and Astrakhan.
Address: 1 Paveletskaya Ploshchad (See Map)
Phone: 235-0522/6807/1920/4109
Metro: Paveletskaya

Rizhsky Station: serves some trains to Latvia.
Address: 79/3 Rizhskaya Ploshchad
Phone: 971-1588
Metro: Rizhskaya

Savyolovsky Station: serves Kostroma, Cherepovets and some trains to Vologda.
Address: Ploshchad Savyolovskogo Vokzala
Phone: 285-9005
Metro: Savyolovskaya

Yaroslavlsky Station: serves Siberia, the Russian Far East, Mongolia and China.
Address: 5 Komsomolskaya Ploshchad (See Map)
Phone: 921-5914/0817, 262-9271
Metro: Komsomolskaya
 
Buying Train Tickets
Recently the Russian Railways (RZD) has significantly improved the services and buying a train ticket is not a hassle any more. Foreigners tend to buy tickets at many travel agencies, which charge a small fee (around 5$). One of those agencies is located two steps away from Sheraton Palace Hotel on 1st Tverskaya-Yamskaya Street. Alternatively, if you speak reasonable Russian, you might want to buy tickets yourself at the railway stations. All Moscow's train stations have ticket booths (called a "kacca").

Moscow - Saint Petersburg Trains
Trains are popular with foreigners shuttling between Moscow and St. Petersburg. The "Red Arrow" and "Aurora" are the most recommended.