Get a map of the subway with station names spelled in Russian and English. Something likethis. Any hotel/hostel should be able to provide one.
Look at the map in advance and find:
1. where you are
2. where you need to go
3. what station is next in your direction (in Cyrillic)
I love Moscow subway not only because it is so beautiful, but also because it is easy to navigate, once you learn how. What's great is that the platforms give you access to both directions, so it is easy to switch if you realize that you are traveling in the wrong direction.
Once you go down the stairs to the platform, keep an eye on the list of stations. At this point all you need to figure out is the Russian spelling of the next station in the your direction.
Let's look at an example.
Say your hotel is on Mayakovskaya station, which is still on the green line. The next station in your direction would be Novokuznetskaya (Новокузнецкая), so that's what you will be looking for on the signs.
"Новокузнецкая" will be listed first, followed by other upcoming stations, and an arrow will show you what side of the platform you need to go to.
Of course, you still have to get out at the right stop. For that I would recommend simple counting of stops. Otherwise, keep an eye on the signs at each station that would announce (in Cyrillic, of course) what station you are at.
Taking a transfer
Now, if you need to take a transfer, you would have to get out at one of the stations that allow a transfer, go to the middle of the platform, and look for a sign with appropriate color. The stations and the directions are color-coded, so you can always see what line you are on and where to go next. In Moscow, transferring from one line to the other often means a lot of walking - corridors are long, but the directions are pretty straightforward. You won't get lost, I promise!
Moscow subway is also very efficient. Time between trains is supposed to be less than 2 min. Each station is equipped with an electronic clock at the top of the tunnel. It shows the time since the previous train left the station, and the next train should come before 2 minutes are up.
And once you are there, check out my favourite stations: Ploshchad Revolyutsii, Komsomolskaya, Kievskaya, and Novoslobodskaya
See also: Ticket Prices In Russia: A Pleasant Surprise!