One of the best ways of exploring Dresden and its surroundings is by bike. There is a good bike path network which makes it safe and easy to travel all over the city and many more attractions are just a short bike ride away from the city centre. One of the nicest bike tours is along the Elberadweg (www.elberadweg.de)
From Dresden, you can go South and you will go past the Dresden castles, the "Blaues Wunder" - the bridge that is close to the beergardens, and after about 1h you will be at the summer palace of August the Strong - Pillnitz. If you are looking for a even longer trip you can go further south until you reach the Saxon Switzerland National Park with its magnificent sandstone sculptures. There are many railway stations close by so if your legs get tired, just take the train back up to Dresden. Alternativelly you can go north to the wineyards of Rabebeul and Coswig, make a detour to castle Moritzburg or go all the way to Meissen. For a bit more action, rent a mountain bike and experience the Dresdner Heide.