The city centre is a traffic headache, especially on weekends. Your best bet is the free Park & Ride lot 3km northwest of the Altstadt, near the airfield on Byk-Gulden-Strasse, where your only outlay will be for a bus ticket.
Local buses cost €2.35 for a single ticket, and day passes are €4.60/7.90 for an individual/family; see www.sw.konstanz.de for timetables. Bus 1 links the Meersburg car-ferry dock with the Altstadt. If you stay in Konstanz for at least two nights, your hotelier will give you a Gästekarte entitling you to free local bus travel.