Newcastle Airport is the the north of the city. so choose somewhere to stay that doesn't involve driving through the city...
Rothbury is an attractive small town 26 miles north along a good road with a reasonable selection of accommodation as well as shops. Cragside, Wallington Hall and Belsay are just a short drive. Alnwick, Bamburgh and the Holy Island ofLindisfarne a bit futher. NOTE that Lindisfarne is reached by tidal causeway so check safe crossing times out here. Don't try and risk crossing on either side as the tide comes in very quickly covering the causeway and foolish motorists regularly get caught out - have a look nat the pictures here...
Another suggestion would be either Corbridge or Hexham to the west. Hexham is larger with more choice of accommodation and shops. Corbridge is very much a large village. These give excellent access to Hadrian's Wall and places like Housesteads, Vindolanda, Chesters. There is also the ruined Aydon Castle to the north of Corbridge.
However all of the different places liisted can easily be done in a day from either Rothbury or Corbridge/Hexham.
Have a look at the map here for lots of ideas of things to do and see in Northumberland. Follow the links to text pages with information and lots of pictures.