The best way to get around Moscow is definitely by Metro. It is so cheap (400rub for 60 trips) and extremely efficient. At busy times there is a metro every 40 seconds and even late at night there is a metro every 5-6 mins. You can pretty much travel anywhere in the city on the metro.
However, the real beauty of the metro system is not the cost of the efficiency. It is the stations themselves. The stations on the brown line (circle line) are decorated so beautifully with stained glass, chandeliers, statues and mosaics. There are no two stations the same.
A great way of seeing these stations is to do a tour of the circle line. Getting off at each station will allow you to see the wonder of these stations.
Taking photos on the metro is no longer allowed due to terrorist attacks a few years ago, however, tourists taking photos of the stained glass and mosaics is still a common sight. Just don't get caught.
I highly recommend a tour of the metro to see transportation system that Russians are so proud of.