The best way for getting to Tübingen is from Stuttgart, where the closest airport is located. Regular buses leave from Stuttgart Airport/Messe very frequently (e.g., lines 828, X82). Alternatively, you may go to Stuttgart Main Train Station, and take a local train from there.
A very useful App, also for moving inside Tübingen, is DB Navigator (Apple Store, Play Store).
In addition to the beautiful old town, with the Rathaus and the Alte Aula of the University, Tübingen's main attractions is the Hohentübingen Castle, where you will also find the University Museum with permanent exhibitions on Ancient Cultures. Just outside the town is instead the Bebenhausen Circestian Monastery from the 12th century. Coming back from there you may also enjoy a walk through the Schönbuch Nature Reserve. Other museums and sights in town: