Saxony has excellent public transport, and if you plan to travel all over the state, the Sachsen-Ticket is valid for unlimited 2nd-class travel on any regional Deutsche Bahn and many privately run trains. The cost is €24 for the first person, and €7 each for up to four additional people travelling together. There is no charge for bicycles.
You can buy tickets online and in stations from vending machines or ticket counters. Buses supplement train routes, and provision for cyclists is excellent across the state. Longer train journeys can often be purchased online with big discounts, if you book at least a few days beforehand and are flexible with travel times.