If you're going to be in the city itself, (the main shopping street), you'll find most of the big chains have very poor selections. There is one store - I'm pretty sure it's Sport-Scheck - that has the best selection of tennis stuff (frames, clothing).
Kronenstr. 3, 70173 Stuttgart (at the station end, off Koenigstrasse)
Not much in terms of strings ANYWHERE in the chains - you'd need a specialist for that and I'm not aware of any in the Stuttgart area.
TW is about a 3 hour drive away from Stuttgart through the Black Forest.
have fun. If you're coming soon, bring something warm. It was minus 18 when I woke up this morning!
"A closed mouth gathers no feet"