A friend an I are going on vacation in Oxford (UK) next week, with an option to stay longer and travel through England. What places should we visit/things should we do under all circumstances?

Our current plan is to stay in Oxford for three days, and "go with the flow" for the next four. In Oxford, we have a few plans, but still a lot of free time. For the remainder, we have no fixed plans other than to do the WB studio tour in London. Mobility wise we are pretty much entirely free, though ideally public transportation as opposed to renting a car would be good.

Do you have any suggestions for what we should do in those free days?