I'm from Eaglescliffe. As another pp has said, lots of the new houses are quite a way out from the high street, but they are nearer Yarm train station with direct trains to York and Middlesbrough, and there are good state schools up that end too.
I prefer Eaglescliffe as its near Preston Park, nice riverside walk, easy to walk to the excellent high school here, Durham Lane Primary is also excellent. Two train stations give access to the coast and Darlington and Middlesbrough.
Not really any parts to avoid, but be aware that houses on the older estates (70/80's builds) are desirable and don't come up very often, so if you're relocating and want to move to the more established estates, you may find it easier to rent and wait for a house to come up.
Depending where you live now, the traffic issue is a bit overblown by the locals, yes it's busy in Yarm and parking is tricky (which is why being in walking distance is great), but even when it's rush hour you'll get through it in 15-20 mins usually, compared to city congestion it really isn't that bad! There is the usual small town resistance to new housing developments but the developments are typical for many areas now.
It's a little gem of a place and we love it, great access to countryside and coast. But the surrounding towns and cities are not the best, Stockton is very run down.