Melbourne is well-endowed with myriads of public libraries replete with a huge array of books, magazines, dvds etc. Free wifi, free internet access, community-based events, kid's activities are some of the other perks of public libraries.
The State Library of Victoria is reputed for its elegant architecture, huge collection of books, computer access, free film screenings from May to October every year, high tea and a horde of other activities. All these for free!
The State Library is pretty... but if you actually want to borrow books, head to the Melbourne City Library. It's hidden away on Flinders Lane, but is worth finding. A very extensive fiction collection.