Registering at the Municipality Office
Withing your first week you should sign up at the Bürgerzentrum and let them know that you live here now. You have to bring your passport (and those of your kids if you have any), your rental contract, and your work contract (in case you do not have your work contract yet contact T. Ehlers for a temporary sheet).
Detailed information for signing up can be found here.
The Bürgerzentrum is in the city center, find the location and opening hours here.
After signing-up you usually get a small booklet with lots of coupons that give you free coffes, reduced entry fees for museums and lots of two-for-one dinner options in the Tübingen area. Hooray!