If you don't want to walk over to Zgorzelec but catch a bus then you have to keep in mind, that your Sachsen ticket is not valid on this bus but you will have to buy one. You find a stop for the border bus e.g. to the left of the post office in Goerlitz on Postplatz.
If you come from Goerlitz by car then you need to take the border crossing over the Pope John Paul II bridge. There are no controls anymore but take a look at the speed limits that are valid in Poland (photo no. 2).
Zgorzelec has two train stations. The Hp. Zgorzelec Miasto is the main one. In therory you can catch a train from Goerlitz to Zgorzelec but it's not that convenient and they don't run very often.
There is a train from Goerlitz to Wroclaw stopping in Zgorzelec. So maybe if you come from within Poland then getting off here may be an option and then you could walk to Germany.
From the old town of Görlitz and from the Neiße bike trail pedestrians and cyclists can cross the river on the new Altstadtbrücke (Old Town Bridge). Neißstraße leads down from Untermarkt to the head of the bridge.
This new bridge has been built in a historical location. Already in the middle ages a wooden bridge connected Görlitz with the suburb on the eastern river bank. In the 16th century it was substituted by a stone bridge. Damaged and rebuilt several times, the bridge was finally destroyed by soldiers of the German Wehrmacht at the end of World War II. There was no immediate need to install a new bridge here, as the new Stadtbrücke further upstream served for traffic between DDR Görlitz and Polish Zgorzelec.
Only in 2004 the new bridge has been completed. It has become a symbol of modern Europe growing together.
Cars have to use the bridge further upstream from the old town, which used to be known as Stadtbrücke. In 2006 it has been renamed after Pope John Paul II. This border crossing has been in use already in DDR times. There is still a border control or customs station, which also seemed manned, but thanks to Schengen there are no controls any more.
This is the only crossing for cars between central Görlitz and Zgorzelec. Of course it can also be used by pedestrians and cyclists.