Since you are going to Dreseden you will be pretty close to Berlin. I think that Berlin is one of the best cities to visit in Europe. There is so much to see and do in the town between the World War II sites, the various culture areas, and incredible night life. It is also interesting to see how different East and West Berlin still are.
Another place to maybe go is Munich. Munich is a great town, but might slightly be out of the way for you. In the town you have the Beer Halls and also Englischer Gardens which is a huge park in the middle of town. It is a very good city to do your own bike tour through. Plus you will be there right before Oktoberfest, so you could see them setting up for that big event.