ditto all above but some extras
south:
hadrians wall walk steel rigg to housesteads fort (can get bus to either end)
newcastle – shopping/nightlife/cafes etc
coast:
alnwick gardens are nice and the tree house restaurant is great
holy island is pretty cool, good walks on far side but can be busy parking/dodgy with tide times
ross sands is super quiet with excellent views of holy island
inland:
rothbury – walks around simonside are good
generally the remoteness and lack of people is the main attraction 🙂
if you are heading from the south the A68/Rothbury route makes a more pleasant journey than driving up the A1 through Tyneside imo. bonus is hamsterley is just off the A68 in county durham and you’re closer to the hadrians wall stuff on the way… also less traffic