You can find an overview of all regional cross-border connections in the brochure of the regional transportation agency VVO (unfortunately only in German). You can download it here or get it at the service centers of VVO.
The Euroregion Elbe/Labe is part of the area of the regional transportation agency Verkehrsverbund Oberelbe (VVO). This area is divided in several zones. Ticket prices depend on how much zones you travel through. Cross-border transport doesn't follow this zone model, but has seperate tariffs per line.
For day trips cross-border the Elbe-Labe-Ticket of VVO and the Ústí region is recommended. It's a day ticket for single users or groups and valid in all regional public transport in the Euroregion Elbe/Labe.
When travelling only by train or outside of the area of Euroregion Elbe/Labe, the Sachsen-Böhmen-Ticket from Deutsche Bahn is a good choice. All regional trains in Saxony and northern Czechia can be used as well as long distance trains of the Czech side.