Don't listen to Stuttgart-bashing, its a nice city! There is a beautiful Zoo/botanical garden called
Wilhelma which is always a favorite for families--I think it gets quite crowded on the weekends though. Wilhelma is at one end of a big park area that goes all the way to the center of the city and I think there is also a natural history museum somewhere in the park (Löwentor?). If your kids are old enough to like cars, Mercedes has a great
car museum, apparently its closed right now as it is moving into a bigger and better building but it will re-open May 20, also Porsche has a smaller museum.
If you like to wander around castle ruins (not castles--ruins!), there are several nice ones within easy driving distance of Stuttgart. When we lived there we liked to go to the following places--we were without kids at the time, but often saw people with children at all three places:
Hohenteck (near Kirchheim) -- here there is already a nice view where you drive to park, if you want the full experience its a bit of a climb and there is a beer garden/castle at the top. Lots of people fly remote-control model airplanes here which can be fun to watch
Hohenneuffen -- a fairly easy walk up hill through the woods and there is a sit-down restaurant and a beer-garden at the top. In the summer they have lots of events, though, like medieval fairs, concerts, etc., so its worth checking in advance if something is on.
Hohenurach -- steep walk, no beer garden, but really nice ruins and a great view. And there is also a waterfall within walking distance of the ruins and Bad Urach has a nice town square with a couple of nice cafes for coffee and cake after your day of hiking.
There's also a thread about
life in Stuttgart.