Photo: Björn Appel, CC BY-SA 3.0 via Wikimedia Commons
A small modern art museum on one of the hills.
Kunsthalle Tübingen is an art museum in the university town of Tübingen, Baden-Württemberg, Germany.
© OpenStreetMap contributors
About 6 km north of Tübingen, right in the Schönbuch forest reservation, is the old cistercian monastery (founded 1187 AD) of with a small village next to it, especially nice to visit at night when part of the monastery is illuminated.
Stadtmuseum
The old city centre full of crooked half-timbered houses, small alleys, cobblestone roadwork and some nice old churches.
The Schloss, or castle, can be a fun place to wander around. Look at the intricately carved gate at the entrance. Skip most of the centre courtyard and head up the stairs on the back side of the courtyard. This leads to a beautiful garden/courtyard, with some beautiful views from atop the wall to the left. Alternatively, go through the tunnel on the backside of the main courtyard that will take you through the backside of the castle, a much more medieval and beautiful part of the Schloss. At certain times, there are archers shooting in the eastern moat (the "Hasengraben") of the castle. From back here there is a path that will take you along the wall of the castle and through the woods, ending up on Haaggasse very near the Marktplatz.
Hölderlinturm
Laboratory dedicated to the cradles of biochemistry and housed in the old walls of Schloss Hohentübingen, in its close proximity to the Ancient Culture Museum.