I grew up in Stuttgart and still live in a town near Stuttgart.
The international tourists that visit Stuttgart mainly go to the Mercedes museum and the Porsche museum:
https://www.mercedes-benz.de/content/ge ... ation.html
Another top place for international tourists is the town Heidelberg with it's historic center about 120km away from Stuttgart. An alternative to Heidelberg is, as already mentioned by other forum members, the town Tübingen, South of Stuttgart and only 33km away from the airport.
There are some other nice places to see in and around Stuttgart...
You can enjoy the panorama on top of the "Fernsehturm":
or visit the "Schloss Solitude":
and from there do a small hike through a nice forest to a lake area and the "Bärenschlössle":
There are also some historic museums and art museums in the city of Stuttgart.
But in my view these sites and places are no top destinations and only valid for a longer stay in Germany.
For a visit to Munich I agree with Simon, and you should have at least 2-3 days. So I would either spend more time in Munich, or if time is short then skip Munich and go directly to Austria.
If you rent a car anyway, then isn't it an option to rent it at the airport in Stuttgart, drive to Tübingen or Heidelberg, and then drive to Austria without taking a train to Munich? On the way to Austria you could make a stop at the "Bodensee" (Lake Constance) and the nice city of Konstanz:
On your way to Austria you could also drive through the "Schwarzwald" (Black Forest), but if you plan to do some hikes in the mountains of Austria, then this might not be an option in your case.
I have been to Austria many times, basically for skiing, but also for sight-seeing in Wien (Vienna), Salzburg and Innsbruck, but I think forum members from Austria can give better hints about places to visit there.