Earlier, I posted some ideas taken from a guide available from the Tourist Information, but you may also want to:
Shop – http://www.inyourpocket.com/germany/stuttgart/shopping– try the pedestrianised Konigstrasse & also go past the Stiftskirche and turn left to find the designer area & Markt Platz -here you’ll find the Wochen Markt on Tuesdays, Thursday & Saturdays, and Stuttgart’s equivalent of Selfridges/Harvey Nicks is Breuninger towards the Rathaus U-Bahn stop on the Markt Platz http://www.breuninger.com/. Be aware that shops don’t open until 10am, but do open until 6 or 8pm most weekdays & Saturdays. However, there’s no shopping on Sundays – all the shops are closed! To keep the kids happy & your pocket empty, try Kurtz Spielwaren (also on the Markt Platz, opposite the Rathaus) or MyToys (Hauptbahnhof end of Konigstrasse) or the 1st floor (toys) or 2nd floor (Wii/DS/etc games) of Müller (also Konigstrasse, but away from Hauptbahnhof & towards Schlossplatz).
Ice skate – Eiswelt http://www.stuttgart.de/en/eiswelt (Ice World) is very near the Fernsehturm and offers iceskating sessions for €5 for Adults, €2.50 (12 & under)/€3 (13 & over), skate hire is free but deposit of €1 required.
Swim – the MineralBad in Bad Cannstatt is probably lovely for Adults, but kids aren’t usually allowed – try Leuze with it’s special kid & baby zones instead: http://www.stuttgart.de/kbb2/menu/187535