If you're coming in from an overseas flight, you'll likely fly into either Domodedovo or Sheremetevo-2; there are also additional airports to handle travel to domestic destinations and the ex-Soviet states. There's a network of comfy-enough buses that run to places within about a 700km (435mi) radius of Moscow. The city also has 9 main rail stations, and you can jump on trains to most parts of Russia and Europe as well as China and Mongolia.
Moscow's metro is the most common, efficient and wondrous way to traverse the city: many stations are fine works of art in themselves. Buses, trams and trolleybuses fill in the gaps. Catching a taxi is simplicity itself - just stick out an arm - but bear in mind that many private cars cruise as taxis and that even official taxis are often not metered: negotiate the fare beforehand.