Both the Staatsgalerie and the new modern collection are actually quite interesting not just for the exhibits but also architecturally. Also, if you are into modern architecture and don't get in too late, there are guided tours to the Weissenhofsiedlung which is a major landmark of Bauhaus architecture at 3 pm Saturdays (no link at hand but google will get you there). No reservation necesssary in theory but a very good idea indeed in practice. And as argued in a previous post, the Stuttgart opera is definitely worth checking out (but not easy to get tickets for!)

Anyone there on a weekday and interested in shopping, don't overlook Metzingen, the nearby hometown of Hugo Boss and the (quality) outlet shopping capital of Southern Germany. A very short drive from STR and a favorite destination of many flight crews (apparently including flight crews into FRA on many occasions, even though it is a much longer drive from there). Lots of options for women as well.