It's really dependant on how much time you have and what your intentions are, both Peterlee and Eshott are the closest to Newcastle, but still a good 30 mins to an hour away on Public Transport.
Flying into Newcastle is relatively easy, and there is a Metro direct from the terminal to Newcastle City. If I recall landing fees are around 15-20 pounds, I don't think it could do any harm to call the GA Handling at EGNT(Samson +44 (0) 191 2864156) to see if you can negotiate something.
I find Newcastle ATC very accomodating to GA and will always try to squeeze you between commercial aircraft.
On the other hand, the great thing about Eshott is that it's fairly close to Alnwick, which is a really nice place, whilst some of the small coastal villages to the East are also great for a nice day out.