There are plenty of pubs around Durham catering to a wide range of tastes. Here is a short list of the more traditional style pubs where you can typically get a good range of traditional-style British ales as well as a variety of Scotch whiskies to aid in the evening's scientific debates.
Durham has a wealth of gentle countryside surrounding the City, which makes a great setting for exploring, both walking and running. A good start is usually to follow the river bank paths, but if you'd like any suggestions, speak to one of the locals or have a look at some of the suggested walks by Durham Pointers or Durham Cathedral. Durham Mystery Treasure Trails also offer a fun way to explore the city (purchase required). Have a look at the This is Durham website for more ideas.