I don't know Oxford too well yet as I seem to spend most of my time working. I'm making a big effort 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 of Art & Archaeology
The Ashmolean is Britain's oldest Museum founded in 1683. There are so many exhibits you will probably not fit them all into one day. I recommend the Egyptology section as they have a lovely collection of Egyptian artifacts here, so that's a good start. Skip the restaurant as it is quite pricey and I give the cream teas a very poor 1 out of 10.

Christ Church Meadow and the Isis (Thames) towpath
This is my favourite place in Oxford. Walk down Rose Lane (near the Botanic Gardens) and enter via a Gate. For the nicest (and longest) walk keep to the left path and follow it along the Cherwell until it meets the Isis (Thames) keep an eye out for Ents you will see Treebeard just before the Cherwell meets the Thames, walk along the towpath and up through the Avenue towards Christ Church College and exit at the St. Aldates Gate. You will see punts, rowing teams, ducks, geese, swans, the herd grazing in the Meadow, Magdelan College Tower and a lovely view of Oxford's Dreaming Spires. I love this walk and it is so hard to believe that such a tranquil place exists in the heart of a city. Take a picnic and a book, it's very easy to spend the entire day here.

The Pitt Rivers Museum