To be honest you will struggle with public transport on the moors. We have the moors bus but its not that great. You would be best off in the northern parts like Glaisdale, Egton,Lealholm, Danby etc as they are connected to Whitby via the railway but also the moors bus.
The Postgate at Egton Bridge is probably the best pub around but cant think of a campsite. Im sure there will be one in Egton Bridge though.
The walking is endless really, particularly in the summer as you could walk to any one of these villages and get the train back.
Rosedale has a good campsite with 2 pubs (3 again soon) in the village. There is a bus that goes through Rosedale but not very often. Your best bet would be a taxi from Pickering.