Zittau has pretty good train connections. Trains run frequently from/to Dresden, (Cottbus-) Görlitz and the Czech city Liberec. If you can, take a Regional Express (RE) which is much faster than the Regionalbahn (RB) - and doesn't stop at every tiny village.
The Sachsen Ticket is good for this train trip from Dresden. Coming from Görlitz you would buy a ticket of the regional transportation network ZVON. The ZVON day ticket for a single traveller is 10 Euro, for a group of up to five people travelling together it is 20 Euro. Please note that for the same price you can also buy the EURO-Neisse-Ticket which is valid in the neighbouring regions of Poland and the Czech Republic (e.g. beyond Liberec into the Jizera mountains). And of course any ZVON ticket is valid on the buses, too, so if you want to do a short excursion into the Zittau mountains your best option is indeed a day ticket.
One of the nicest way to do the side trip to the mountains south of Zittau is to ride the narrow-gauge steam railway, which operates since 1890. This steam train starts right next to the main train station Zittau where the trains from/to Dresden, Görlitz and Liberec arrive/depart.
The steam train runs in a half circle around the eastern part of Zittau, close to the German border, to Bertsdorf where the railroad splits - one goes to Kurort Oybin and the other to Kurort Jonsdorf. This allows you to take the steam train to one of them, e.g. Jonsdorf, explore the village and hike the two hours to Oybin, see the sights there and take the steam train back to Zittau from there.
Some trains have bistro cars where they offer simple food/beverages, some have open-air cars which are fun to ride.
The steam train company acknowledges the day- and weekly tickets of the regional transportation network ZVON but you have to pay a steam supplement. For example, you can ride the steam train with a ZVON day ticket (10 Euro) but have to pay a 4 Euro supplement (valid all day). For shorter train rides you need to buy individual single tickets from the steam train company (available on the train or in their small station).
From Dresden you can take a regional train to Zittau which takes just under two hours. We used the "Sachsen ticket" for the day for 27 Euro (2008). Up to 5 people can travel on that ticket and the good thing is that it is also valid on public transport in the towns ie for trams etc in Dresden, Goerlitz and so on.
We actually took such a regional train to Goerlitz first, then a Connex train to Zittau and travelled from Zittau to Dresden by such a train in the dark.
This line from Cottbus via Goerlitz to Zittau is called "Lausitz Bahn" and it's run by the Connex company. Your Sachsen day ticket is valid on this line.
The line travels along the German/Polish border and actually stops in Poland. Another feature is that you can buy inexpensive drinks and snacks from the conductor while riding on the train. We only realised this too late. The train ride from Goerlitz to Zittau takes around 40 minutes.
I usually come from Autobahn 4 from Dresden and leave at exit 90 (Weißwasser) - do not take exit 89 (Hoyerswerda) as the road signs suggest because this road leads through Bautzen and the other is just in the citiy's peripherie. Follow the federal road B6 towards Löbau and turn there on B176 towards "Herrnhut". from here the road signs guide you to Zittau.