Take a walk through the Rosensteingarten and then up to the Killesberg. Killesberg provides a perfect view over Stuttgart. Wilhelma, a zoo founded by King William I in the 19th century and completely built in Arabian architecture is a real exotic spot in the middle of ancient buildings and places. The botanical gardens of the Wilhelma too are just marvelous. The botanical garden has different themes over the year but if you like the lotus flowers in blossom make the trip in May or June. If you are tired, relax in one of the many public bathes like Leuze's or the Cannstatt Bath.
A good point to get an overview of the City is the restaurant on top of the TV tower. It is expensive, but you will be rewarded with a splendid view of the entire Neckar valley and the city.
The Staatsgallerie is not to be missed and contains collections of medieval art up to modern art from Max Liebermann to Picasso.
Take a stroll thru the busy Koenigstrasse - the pedestrian zone. It will give you a time for shopping and also the feeling to be in a real metropolis. In the Calwer Street, a small side street, there are many nice and snug restaurants, pubs and also some clubs. A famous place is the Calwer Microbrew. But it is usually too crowded after 5 p.m. The Tuebinger Street is the haunt of students.