The definitely most economical and a surprisingly convenient/pleasant mode of transportation to St. Marienthal is cycling. The cycling trail along the river Neiße is well marked and maintained and runs from (Liberec-) Zittau to Görlitz (- Baltic coast). The scenery is lovely, the views of the mountains (Jizera e.g.) are beautiful. Plenty of restaurants and B&Bs along the way - just in case you're tired or starving.
The abbey is right on the cycling trail - you cross the grounds when cycling on the trail. The distance from either Zittau or Görlitz is 20 km - no sweat!
Another option to get to the abbey by public transportation is the bus that runs along the main road B99 Görlitz - Zittau.
The problem with the bus is that it runs infrequently (roughly every 2 hours) and only on school days. Expect hords of school kids in the bus, too! The bus starts sometimes at the train station in Görlitz but mostly in Görlitz-Weinhübel which is accessible by tram #3 from Görlitz centre. Your best bet is the train station Hagenwerder, though, which you reach by the trains mentioned in the previous tip. There you change to the bus.
The advantage of bus vs. train is that the bus takes you close to the abbey. It's only a 5-10 minutes walk from the bus stop to the abbey. Fares are the same like for the train (same transportation network - ZVON).
Getting to the abbey by public transportation is a bit tricky and inconvenient. There are trains, operated by the ODEG (not Deutsche Bahn!), that run Görlitz - Zittau, a quite scenic route that partially runs on Polish ground. The closest stop to the abbey is Krzewina Zgorzelecka, just across the border from the town Ostritz (2 minutes) and from there you either catch a bus to the abbey or (as they do not run frequently) just walk along the Neiße river (cycling and walking path) - about 25 minutes.
Sachsen-Ticket and ZVON (the regional Transportation network) tickets are valid. One way single trip from Görlitz in 2012 is 3.40 Euro.
The Abbey St. Marienthal is located - typical for a Cistercian convent - in the countryside, actually some km from the next (small) town, Ostritz, and half way between the next bigger towns Görlitz and Zittau. Thus, driving is the most convenient and fastest way to get to the abbey.
Parking (free) is right off the main road Görlitz - Zittau. The parking lot is quite large, except on special occasions like wine festival or other big events there should no problem finding a spot for your car. From there it is a 5 minutes walk down to the Abbey.
There is another, very small parking down by the restaurant Klosterschenke (see picture) but it is reserved for guests of the restaurant only.