I don't think there are any buses between Stanhope and Barnard Castle (Weardale to Teesdale), but if you do get a taxi there are lots of good views on the way -
https://goo.gl/maps/b2PZXbTbBYn
There's normally a lot to see at the Bowes museum if you want to take it all in too.
Bishop Auckland West is a platform for the Weardale Railway (i.e. not a station building like Stanhope). It's next to Bishop Auckland station, but there isn't a direct link between the two. I went earlier this year and it's across the B&Q car park, down the side of the Royal Mail sorting office and through a gate. If you go in Bishop Trains at Bishop Auckland station I find they are very helpful and will point you the right way. Morrisons is probably closest for toilets before setting off...