The easiest and fastest way to get to Sergiev Posad is to rent a car or hire a taxi and drive around 70 km along the highway Moscow - Yaroslavl.
Train is another option. There are several daily trains between Moscow and Yaroslavl that stop in Sergiev Posad. Travel time is about 1 hour.
Buses from Moscow to Sergiev Posad depart from VDNKh bus station which is located by the metro station of the same name in Moscow. Travel time is about 2-3 hours.
In Sergiev Posad both train and bus stations are located in 2 km from the Monastery. Bus/taxi service is available.