Public transport is pretty cheap in Moscow, and you can get around by metro (subway), bus, tram, taxi...The most popular and convenient method of travel is the metro.  Tickets are very cheap and are purchased at the metro stations, at the window labeled with the word "KACCA".  You can buy tickets for various numbers of journeys - from 1 to 60 journeys.  Once you have purchased your ticket you go through the turnstiles and then down to the platform.  The metro trains arrive around every 2 minutes during peak times and the time gap between trains extends to about 10-15 minutes later at night.  While being a convenient method of travel, the metro does get very crowded, especially at rush hour, and you'll often find yourself packed in like sardines!  That said, the metro is very efficient and many of the stations are works of art in themselves.  Some of the best known ones include Komsomolskaya - with its chandeliers, Kievskaya, with its beautiful artwork, and Mayakovskaya, with its marble decor and mosaics on the ceiling.  For more information, see the following TripAdvisor Inside Pages How to Use the Metro; Moscow Metro: Underground Palace

Other methods of transport include trams and buses.  You can normally buy tickets for these in advance, at metro stations and kiosks.  Some metro stations do not sell them however and it is best to try kiosks adjacent to bus stops and ask for 'avtoboos beelyet'  and they can be bought from the driver for a couple of roubles more.  Another option is to get a taxi.  There are taxi firms in Moscow which you may wish to use when going a longer distance (e.g from the city centre to the airport) but to go a shorter distance it's common to hail an ordinary car in the street.  While in Moscow this is a common way of getting around, caution should still be exercised - avoid hitch hiking alone or at night.  It's best to agree a fare in advance, but you can usually go a short distance quite cheaply.