Take a train in a Rizhskiy direction from Moscow. The best places to change are Dmitrovskaya and Tushinskaya metro stations, if you try Rizhskiy Train Station, you'll find, that a large number of trains are actually passing only Rzhevskaya station on the other side of the square (some 10-15 minutes on foot), which is a bit confusing, unless you know directions and timetable beforehand. Time varies from 1 hour 15 minutes to 40 minutes depending on where you start, and if you using express (from Dmitrovskaya: 10.24 on weekends and 19.15 daily) or not. Tickets around 100 roubles.
Buses to Istra run from Tushinkaya metro station and prone to rather heavy jams between MKAD and Krasnogorsk, and inside Krasnogorsk. If you go to Istra by car, it's better to start via the new M9 highway (getting out of Moscow via new Krasnopresnenskii prospekt, Zhivopisnii bridge and Serebryanoborskii tunnel), then switch to A107 (first betonka), and then to old Volokolamskii highway.
You can walk about 2 km from either Istra or New Ierusalem stations to the monastery. If you want to take bus - №4 will take you there from both stations, otherwise the scheme of buses is a bit confusing. You can take №№26,32,33,34,40,46,48 from Istra station to museum as well, and from New Ierusalem station every bus bound either to Istra station or Uritskogo square; however, a lot of buses take different routes (mostly just shortened ones) even with the same number; so a walk may be a fair variant.