I love libraries and we are lucky in Graz to have several great options.
Stadtbibliothek Graz is the city’s public library system. There are several branches around Graz, a library bus that visits the outskirts, or you can also chose to have your books delivered by post. A library card is free for children and 15 € for adults. They have several shelves of English fiction (and other languages) for both adults and children. Plus they now offer 1,000 digital English e-books! There is also a great selection of board games to borrow. I can also recommend their kids story times, called Labuka which includes a short craft after the story (also offered in English and other languages). Plus they have a special media library (next to the main Zanklhof libary) with a huge selection of DVDS and music CDS. Also at the media library is a new lending service – with lots of sports equipment, household items, work tools, cameras etc to borrow (all for free with your library card!).
You can find more library information (lots of great events too) and the registration form in English here. The librarians at the central library Zanklhof and also at Graz Nord in Geidorf speak good English if you need help.
Steiermärkische Landesbibliothek is in the Joanneum Quarter, in the underground welcome center between the Natural History museum and the Neue Galerie. They also have good English fiction selection and a kids library. They have a large quiet reading / study room and big selection of magazines (including English ones such as the New Yorker). Library cards are 7€.
The English American studies department library at the Karl Franzens university has a large collection of English fiction and non-fiction. You can get a library card free of charge from the main university library.