My ex-bird lived there, so I've spent a fair bit of time in Stuttgart.
As others have said the centre of town is a bit non-descript but there's a nice palace and gardens in the centre of town. The Mercedes Benz and Porsche Museums are worth visiting if you're into your cars. Other things to do include going up the TV Tower (Fernsehturm) which is directly above Stuttgarter Kickers' stadium. There's also a decent zoo in Stuttgart.
I can't see what dates you are looking at going, but Fruhlingsfest is coming up which is the Spring version of Oktoberfest. I've done it a few times, and the beer tents are superb! The beer festival stuff takes place in a part of Stuttgart called Canstatter Wasen, which is beside VFB Stuttgart's stadium.
Outside of Stuttgart, Esslingen is a really pleasant wee town 20 mins on the train from Stuttgart. It has a nice historic old town, and a castle high above the town which you can walk up to.