Additional question: Where are you staying?

If not at a hotel near Stuttgart Main Station then make sure to directly book your train ticket to your final S-Bahn station. Urban trains (= S-Bahn) are included if printed on the ticket as final destination. If you buy only until Stuttgart Hbf then you'd have to buy 4 single tickets (or day tickets) from there onwards.

You can buy a train ticket from FRA to any station seen on http://www.vvs.de/download/SBahn_Liniennetz.pdf at no additional fee (only S-Bahn; U-Bahn is not operated by Deutsche Bahn and therefore always costs extra).