Badeparadies Eiswiese
One of Göttingen's public swimming pools, Badeparadies Eiswiese (http://www.badeparadies.d...), is a roughly 20-minute walk south of the city or a ten minute bus ride on line 4 or 14. But if you enjoy the water, they offer a plethora of swimming pools indoor, outdoor, large, small, hot, cold, saunas, jacuzzis, and solaria.
Check the city calendar (http://kalender.goettinge...) for local events, or pick up a schedule in the Tourist Office in the old town hall. The symphony orchestra is impressive for such a small town.
The Göttingen forest follow Herzberger Landstr. east is full of pleasant hiking trails, as well as the Bismarck Tower, which you can climb for a gorgeous view in the summer though you may need to ask directions, since the tower is hard to find.