Meersburg is the obvious first choice — a hillside town on the opposite shore of the lake, fifteen minutes by car ferry and worth every minute. The medieval castle (the oldest inhabited castle in Germany, claimed) dominates from above, the Altstadt is genuinely beautiful without being theme-park smooth, and the vineyards running down to the waterfront produce some of Baden's better Pinot Noir. Take the ferry rather than driving round the lake.
Mainau Island can be a day trip or a half-day from Konstanz. The subtropical gardens, the butterfly house, the rose terraces — it's described in full in the Things to Do section, but it earns its own day if you take the time to walk the whole island slowly.
Stein am Rhein is a Swiss town about twenty kilometres southwest of Konstanz, where the Rhine leaves the lake. The main square (Rathausplatz) has some of the most spectacular painted facades in Central Europe — entire building fronts covered in frescoes depicting local legends and biblical scenes. The town is small and crowds thinly on weekdays. By car it's under half an hour; by bus it takes a little longer. Cross the border into Switzerland with your passport.
Bregenz is the Austrian shore of the lake, about an hour east of Konstanz by ferry or fifty minutes by train. The Pfänder mountain above the town is accessible by cable car and gives panoramic views of the entire Bodensee. Bregenz is also the home of the Bregenzer Festspiele — a summer opera festival staged on a floating stage on the lake, one of the more extraordinary theatrical settings in Europe. Performances run July and August and tickets sell out months in advance.
Friedrichshafen, directly across the lake from Konstanz, has the Zeppelin Museum — the airship history of the Bodensee region is substantial, and the museum does it justice. The ferry takes about an hour.