S-Bahn lines S2 and S3 run from the Flughafen to the Hauptbahnhof in 27minutes.
An excellent starting point is the english-language Welcome page of the VVS website (the VVS is the entity which runs public transit in the metropolitan Stuttgart area):
http://www.vvs.de/en/index.php
The main site is at:
http://www.vvs.de
Use the Deutsche Bahn site to construct your complete trip from the airport to your final destination - transfer times will be listed:
http://www.bahn.de/i/view/GBR/en/index.shtml