Hello! Indith's done a great job there (and the village where she lives is as nice as it sounds - I run out that way quite often and it's lovely down by the river).
I live in Durham, and have to say I love it. I've lived in cities in the South of England and it's a totally different experience, so I wouldn't necessarily discount the idea of living in the city, especially if you want to send DC to schools in Durham itself anyway.
I don't especially notice the traffic problem for getting into Durham because once you're in it, you can get all over the place without going onto roads at all. I went on a 6 mile run today and crossed 2 roads! If you look on the map, you'll see there's a lot of wooded area to the south and west (where I live). Once in Durham city centre it's pedestrianised (and the castle and cathedral are stunning, it's such a privilege to be able to see a sight like that when just out shopping!)
On to practicalities, I'm on the west side of Durham and in walking distance of Durham School, Choristers and Durham High School for Girls. I've friends who have sent their DC to the two latter schools and speak very highly of both. Our DDs go to the state primary, which is very good. Where we live there is a high preponderance of academics' and doctors' children, and well (without wanting to sound too snobby, which I do, but there we are) it does mean that the local state schools have lots of very academic and driven children, a lot of multi-linguality and multi-culturality going on, and it makes for a very good growing up experience.
The railway link to Newcastle is very good (there are some excellent private schools in Newcastle, too. I used to be a solicitor in N, so knew quite a lot of people who sent their DC to RGS, Central High and Church, and they're all quite different but again, very recommended by those individuals).
I would definitely recommend coming to have a look around. You can easily tell from the house prices what the area might be like. Some of the pit villages are very depressed, so you really want to pick your village carefully! If you wanted proper country (i.e away from village life) there are some stunning properties about, though I wonder how often any of them come onto the market.
Come up to visit, eat cake with us and we'll give you the low down!! I think it's a great place to live though - the quality of life up here is SOOO much better than in the SE, imo.