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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.