Newcastle is the natural base for the eastern half of Hadrian's Wall. Segedunum is on the Metro at Wallsend, and the forts at Chesters, Housesteads and Vindolanda are about an hour west by car, or by the AD122 bus from Hexham in summer.
Closer in, the Metro runs to the beaches at Tynemouth and Cullercoats in under half an hour, and Durham, with its cathedral and castle, is fifteen minutes by train. The Angel of the North and Beamish open-air museum are an easy drive south, and Whitley Bay and South Shields are on the coast either side of the Tyne.
The Northumberland coast is a long but rewarding day out: Bamburgh, the tidal Holy Island, and the boat trips from Seahouses to the Farne Islands.