To bike to the university, take Friedrichstrasse up from Zaehringerplatz. The street name will change to Universitaetsstrasse and lead you directly to the university’s main entrance.
The city of Konstanz provides information (in German) about biking in Konstanz, including how to take bicycles on trains or buses. An overview of bike routes is available in the city biking map (in German).
There are plenty of bicycles for rent in Konstanz: 150 “konrad” rental bikes in addition to 26 “TINK” rental bikes for transporting larger items. You can use any of the 15 “konrad” stations or any of the 13 “TINK” stations to pick up a bike.