When you wander around the "Niederburg" you will find some small winebars. They are pretty old, charming and loved by the locals. People will meet there to have a glass of wine ( or maybe two or three....) Try it.
Konstanz has it's own wine, which can be purchased in wine stores and supermarkets in the region. If you'd like to try it first you can visit one of the wine bars or attend a wine tasting. There a several wine tastings available that you can book as a tourists. Some wine tastings are offered with some snacks too.
Ask me, if you fancy a wine tasting. I will send you more informations.