The Black Forest Schwarzwald in Germany is about 2 hours away by train or 1.5 hours by car.
Other cities and towns on Lake Constance. Especially recommended are Meersburg, Lindau and Bregenz.
Stein am Rhein
See the Rhine Falls in Schaffhausen
Reichenau Island
Neighbouring Swiss city Kreuzlingen has a really nice waterfront with large parks and playgrounds and a small zoo. Simply walk south along the waterfront from Konstanz to get there.
Swiss Alps. In the winter, sign up at one of the local sports stores e.g. Intersport Gruner across from the Lago Center, or Karstadt Sport in the basement of the Lago Center to join a bus day-trip to a ski resort.
