Aww I love James Herriot, I was really sad I missed seeing some of his associated places when I was in the area.
Some National Trust recommends .. I'm way too big of a fan.
Wallington House is GORGEOUS. I think you can get there by coach and it's not too far a drive. It has amazing grounds, literally the Secret Garden is contained within them.
Fountain's Abbey is beautiful. There's nothing more to say without serious hyperbole. Go!
I really liked Hadrian's Wall, but I'm a bit of a classics/archaeology nut. It's a great place to go and think about how short a time a thousand years is just in the scale of things.
Not National Trust, and I never made it (I was sick one free weekend, occupied in Durham the next
) but I heard Alnwick Castle is worth a trip, the gardens have quite the curiosities and you can find enough in town to fill a day trip.
Within Durham, have you been up to the top of the Cathedral? It's scary - but exhilarating. The side chamber where they shot Harry Potter interiors is quite nice too, then you go back and watch the film... again... If you haven't been, the Oriental Museum is quite nice. They have a small, but well-organised collection, mostly East Asian.
There's a really fun wee dress-up box too (okay, I had fun with it
), and they have an Egyptian section also.. with an actual mummy. Plus when the weather is nice the walk along the river is one of those pleasures I dream about having again in my life.