Hi Bradley, Beamish is a must but you could spend days there alone, I have not been but York and Darlington and I think Shildon all have railway museums, there is also the Discovery meuseum in Newcastle which has engineering related things in it. The Laing art gallery in Newcastle is also top quality place all sorts of class items from the building itself to artwork of engineering and shipyards etc, top class local art of all types , glass, ceramics, wood carvings, silversmithing, goldsmithing , Dame Laura Knight exhibition is still on too I think.
As for the Romans you are spoilt for choice in this area from the museum of antiquity s in Newcastle university and all along the wall from Wallsend to the west coast. I would say Vindolanda and Chesters are well worth a visit and give a good over view of the area in Roman times.
Hope you enjoy your visit.
Cheers Mick.