Dresden has a very well developed public transport system. Trams and buses in the city centre are operated by the Dresden Transport Services (DVB)  and regional transport is taken over by the Oberelbe Transport (VVO)

Schedules and informations about the ticket prices are provided on their websites and at the stops.

DVB Network Map 

Tickets can be bought on board the trams (vending machine) and busses (from the driver), at stops (vending machine) and at service points. Riding without a valid ticket will result in a fine. City trains also run in Dresden and the surrounding area. Travel centres in the Dresden Hauptbahnhof (main train station) and in Dresden-Neustadt will provide you with further information.
You may also by tickets via mobile phone (only in German): ticket via mobile phone