The Bodleian Library is the Library of the University of Oxford, so it is a working library. There has been a library at the University since the 1300's. It's free to explore the Old School's Quadrangle and the Exhibition. The architecture is outstanding. A tour of the Library costs £12.
The Park is a great place to visit at any time, but especially during the summer. Visit the "Covered Market" and pick up some food and take a picnic to the park.
I don't know Oxford too well yet and I seem to spend most of my time working. I'm making a big effort this year, 2010, to get out and about more and in particular find out where are the best places to go for free. Here are a few places to get you started:
The Ashmolean Museum
The Pitt Rivers Museum