The Metro is the underground. It is very cheap compared to London. A single ride on it (ie. as far as you like without leaving the station) costs 20 rubles, which is about 40p.
The routes can be confusing, but there aren't as many lines as there are in London, and the maps on the trains themselves are in Russian and English.
The main thing is the same as always in Russia, and that is not to look conspicuous. Most people on the Metro keep quiet, and speaking english might get you unwanted attention.
Look on maps and signs for a blue M.