Oh man I wish you'd said something a few weeks ago, I need to visit Durham but I figured I'd just have a wander round when I'm there in the summer. I could have come too! It's a bit late now since it's today... Ah well. I hope you get chance to read this while you're there.
Durham and Edinburgh tie for my favourite cities in the world, so you'll have an awesome time. What are you looking to study?
So, in Durham: Go to the cathedral. Go to it. I don't care if you don't like cathedrals. Go. It is lovely. Also I believe you can do wax rubbings there, but I don't know if that's still the case.
There is a very pretty river but I think that isn't really an activity, more of an observation to make. Make your way to Millenium Square and the market, I think there's an art gallery there but I might be making it up. The Gala Theatre has a cinema and also a theatre (obviously) and it's nice but not cheap. If you'd been there yesterday there was a Pay What You Like play on, but you missed it.
Have a good time! I'm really quite jealous, I've been sorting uni open days today and I wish I could make it to the Durham one but it's too far and not really sensible when I'm in the city anyway in the summer.