£5.70 single from Durham to Newcastle, typically takes 15 mins. Walk/bus (TEN)/Metro to Eldon Square (Haynarket for the latter) then catch the 685.
Decent places are everywhere, you could do in the city centre (Sainsburys, West Tower St, The Courts) as it's a unique. Another place I'd recommend is London Road as it's one of the main roads for city services as well as the Cumberland Infirmary. My favourite route is the 67/68 from Belle Vue to Upperby. Bus depot if you're interested is around 10min walk from Sainsburys which is near Carlisle Castle.
If you'd like I can go with you, date dependent as it'd be a good little day out for me also (I work in Carlisle but still need a proper bus bash!)
Re Tom, 400 is Enviro 200 operated I'm afraid. Only B10Ms you'll see are Workington's on the likes of the 600 but not guaranteed.
RE: Bus Related Outings and Reviews