Stuttgart ist the Capital City of Baden-Württemberg.
At the S- Bahn station Osterfeld is a parking house were you can get a traveling pass for public transport. This you can use the hole day in the inner city. You only have to pay for it when you go out of the Parkhaus with your car but if you don't have a car you don't have to pay.
Public Library "Stadtbibliothek Stuttgart"
The public library has a computer terminal on the second floor that can be used by everyone for free for about 20 minutes. From Hauptbahnhof (Main Station), you can take metro U5 (Leinfelden), U6 (Fasanenhof), U7 (Ostfildern), U12 (Möhringen/Vaihingen), or U15 (Ruhbank/Heumaden) and exit at the stop Charlottenplatz; or you can get there by foot in about 10 minutes. It's open from 10:00am to 19:00pm on weekdays and from 10:00am to 16:00pm on Saturdays.
See http://www.stuttgart.de/stadtbibliothek for further details (in German only).
There are several WiFi-Hotspots around in bars that are free of charge, but you usually have to order a drink to get a voucher. Some of the spots that are not too far from Hauptbahnhof (Main Station) are listed below:
- barcode, Theodor-Heuss-Straße 30, http://www.barcode-stuttgart.de
- muttermilch, Theodor-Heuss-Straße 23, http://www.muttermilch-stuttgart.de
- suite212, Theodor-Heuss-Straße 15, http://www.suite212.org
Also you can get free wi-fi in every Starbucks, there is no password for an access, it's unlimited and nobody cares if you stay there for hours without buying something.
You can put your luggage in the free lockers at the stattsgalerie Stuttgart. you just have to have one euro for getting using it but you get the euro back when you take your luggage again. The Staatsgallerie is closed on Monday on all other days its open from 10am o'clock to 6 o'clock pm on Thursday till 8pm.