In terms of art and culture, off the top of my head, I came up with:
* The Palace Green area of Durham, which is a UNESCO World Heritage site.
* The Cathedral, where St Bede and St Cuthbert are buried, contains a small museum but is an important historical site in its own right, being widely considered as the greatest Norman building in the UK (possibly even Europe) and one of the best and earliest examples of Romanesque architecture.
* Many University buildings are listed and considered of historical and/or national importance, including Durham Castle (University College).
* Palace Green Library, also listed, played host to the Lindisfarne Gospels and exhibition this year and there is a heritage centre on the Bailey.
* The University itself has two museums, the Oriental Museum and the Archaeology Museum and has Botanic Gardens.
* The City has the DLI museum and Art Gallery and two theatres (one of which is part of the University).
* The University also has one of the largest art collections of any university in the UK. Some colleges will host an artist in residence (including writers and painters) who will run seminars or events for students, staff and members of the public.
Nearby, in County Durham, there is the multi-award winning Beamish Museum at Chester-le-Street and, if you have transport, Washington Old Hall (ancestral home of George Washington), Binchester Roman Fort, the Bowes Museum, Raby Castle and the ruins of Barnard Castle at Barnard Castle and Escomb Church (the oldest complete Saxon church in the UK). Some or all of these may be accessible by public transport as well.
A short train journey (15/20 minutes) and the Sage Concert Hall at Gateshead and theatres and museums of Newcastle are all accessible. Durham is also well connected on the East Coast mainline for a day visit to York, Edinburgh, Harrogate and other major cities (including London!).
Hope that helps!