good that you have a car - that gives you nearly unlimited possibilities. :)

Concerning the lake you already have received some ideas, and biking is another one. Many people do even a complete round around the lake by bike.

You are on the edge of Switzerland, where you can reach a lot of interesting points within a day from Constance. The black forest is not far away - which would like the alps be rather challenging for biking.

And then - if you're interested in culture, on the other side of the lake (north shore) is a famous area for baroque architecture. Very near would be the Birnau, a very nice baroque church but there is much more around.... This area we call "Oberschwaben" btw if this might help you searching, and there is a route to these baroque buildings called the "Oberschwäbische Barockstraße".

hth Bärbel (near Stuttgart)