Hiring a taxi to move around Moscow is rather expensive. But if you are using a car, don’t forget that traffic in Moscow is pretty heavy, and you risk getting stuck in it for a long period of time.
The best way to get anywhere you want is metro. Although Moscow subway does have some disadvantages. It is very much overcrowded. During rush hours it is almost impossible to get on a train.
Good thing about metro is that it’s very pretty, so you can spend some time admiring appearances of metro stations while waiting for the next train.
Of course there are some other means of public transportation: buses, trolleys, and trams. In some places you can use water bus or railway to get to your destination.
Most of the tourist spots are located in the center of Moscow, so you can actually walk around. It might be even more convenient and faster since public transportation is not that extensive in the central part of Moscow.