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 (press your card up against the small circle - it will also show the number of rides left) and then down to the platform.  Or you can now buy a stored value plastic card called "Troika" at metro station. There will be a discount of the fare when you use the card.  This "unified" card can be used in any form of public transportation - Metro or surface, and if used for both in one trip - that is a transfer from one form to another - within 15 minutes or so, offer a significant, further discount. You can either top up the card at the metro counter or, much more preferable to avoid lines, using the machines at the station which accept both coins and notes.  Be reminded that  Since any remaining value stored in the card may not be refunded.  you may want to calculate the amount you would use around the end of journey.  The metro trains arrive around every 2 minutes during peak times and the time gap between trains extends to about 5 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.

One other note: beware the (wildly) swinging glass doors...both entering an exiting, these things can swing back pretty far and hard, so make sure you are ready to "catch" the door if someone has gone through right before you!

Other methods of transport include trams and buses.  While you can normally buy tickets for these in advance, at metro stations where as above you can use the same, unified "Troika" store value card, and kiosks adjacent to (some by an ever decreasing number of) bus stops and ask for 'avtoboos beelyet'  and they can be bought from the driver for a couple of roubles more. There is a powerful tool to plan your route via public transport. It situates at  yandex.maps The interface is in Russian only but it's very simple. 

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), or, for much cheaper fares, there are taxi apps including Uber, YandexTaxi, and Gett. Note that is no longer legal for private ("gypsy") taxis to give you a quick ride as had been the case for decades. But, given the much lower ruble, and use of apps, fares in Moscow are extremely reasonable compared to other major world cities. Also, keep in mind rush hours - taxis may seem convenient at times, but during rush hour can turn a trio you could make via Metro and on foot from 15 minutes into an hour!