Stuttgart sits in a major German wine region. It's well worth a visit to the vineyards in the hills just on the border of town (pictured at left). There is wine tasting, which is perfect at sunset. Note that red wines, and not just the German white Riesling, are available. I always thought of Riesling as a somewhat sweet white wine (good on a hot day) and, while many are, there are some excellent dry ones. You can visit one or more taverns up in the vineyards and enjoy the local wine specialty, beautiful views, and good Swabian food.
For dinner in town, go to Cube (www.cube-restaurant.de), an exceptional modern restaurant set on the top floor of the contemporary art museum (in the pedestrian shopping zone). The food is outstanding, the crowd is trendy, and the views over the main square are appealing. I tried pork filet (and generally I'm not a huge fan of pork) it was outstanding. Also consider Swabian specialties including spaetzel and Swabian ravioli (with beef filling).
Lastly, Stuttgart has its own Oktoberfest. While Munich gets star billing for this, Stuttgart has an impressive set of parties itself. For sure there are fewer foreigners at the Stuttgart Oktoberfest.
Baden-Wurttemberg is an under-appreciated region for Americans. That's somewhat surprising since it was once one of the American-administered zones post WWII. In fact, one factoid is that the Americans encouraged Baden and Wurttemberg, separate regions at the time, to merge so as to have one civil administration rather than two (and thereby save money). The region has a lot to offer with Heidelberg, Baden-Baden, the Black Forest, Stuttgart, and Esslingen. My only regret was not having more time to relax and luxuriate!
For general trip-planning information, see our Germany Travel Guide.