I've never been to Durham but if I did go, I'd do roughly the following:
1. Listen to the ATIS on 136.2 first to get the runway in use and other relevant information, noting the current information letter.
2. Make sure I have some idea where I am.
3. Make an initial call:
"Durham Approach, G-ABCD, inbound."
Durham will reply with something like "G-ABCD, Durham, pass your message".
4. Give a standard response, something like:
"G-ABCD, PA28 inbound to you from XYZ, 5 miles north of Sedgefield Racecource altitude 2,000 ft QNH 1027, information charlie, request joining instructions."
You'll probably be asked to report first at some VRP before being cleared into the zone. Durham Tees has six visual reference points (VRPs) so make sure you know what they are so you can recognise the names and know roughly where to look on your map.
You may at any point be told to standby and/or to remain outside controlled airspace. Give yourself plenty of time by calling in good time.
You may also get some specific questions or instructions in response to your very first call, especially if they have your details already. Don't be put off, just give them the information they want.
The definitive guide is of course CAP413.
As others have said, your instructor should be able to give you the best advice as s/he'll be familiar with the airfield and any local procedures.
Mark