One can easily get from Moscow to Sergiev Posad. Sergiev Posad's transport hub is its busiest area.
Trains leave from the Yaroslavsky Terminal (the Komsomolskaya subway station) every 0.5-1 hour, rarer in lunch time. They take 1.5 hours and about €3 one way. This railroad line goes further through Alexandrov, Rostov the Great, Pereslavl-Zalesky to Yaroslavl. Through a window you can see great inside views of Russian countryside. But note that a commuter train is a mid- and low-class transport full of curious and drunk seatmates. If in a hurry, taxis are available behind the terminal.
A more comfortable transport is bus from the ВДНХ (VDNH) subway station departing every 10-15 min. For about the same price as train, it goes for 1 hour. The bus tickets are sold on board.