Re: bedding: if you end up living in student dorms at WHO (Waldhäuser Ost, the largest cluster of student housing), facility management will sell you (or rather, make you buy) IKEA bedding, sheets, and covers. It'll be the cheap kind, about €20-30 altogether, but absolutely does the job (and you can still get together with friends, rent a car, and go to IKEA for more/nicer stuff later on).

As for "to do": aside from the touristy thing, you could take art classes with university or the VHS (community college type thing: https://www.vhs-tuebingen.de), volunteer with the DAI (Deutsch-Amerikanisches Institut: http://www.dai-tuebingen.de), do sports, hang with friends … and if my friends' experiences are anything to go buy, you'll also be studying a whole lot.

As for sports, uni's program is pretty diverse and cheap -- have a look: this is the list of what they're currently offering (all in German, sorry): http://sport.ifs.uni-tuebingen.de/angebote/aktueller_zeitraum/index.html If you're willing to pay a bit more, there's tons more opportunities for sports outside of uni, too. Even nightlife doesn't have to mean (just) drinking. You can go for all the 'boring and usual' stuff like billards, darts, or bowling; there are still a couple of places that do pub quizzes etc.

Good luck with your applications!