DH is from Northumberland and I have spent a lot of time there. This question pops up a lot on mumsnet so I have drawn up a list of things to do:
Towns
Alnwick, Amble, Berwick upon Tweed, Corbridge, Hexham, Morpeth, Rothbury, Warkworth
Castles
Alnwick Castle and gardens, Aydon Castle, Belsay, Bamburgh, Chillingham, Dunstanburgh, Lindisfarne, Warkworth
Stately homes
Cragside, Seaton Delaval, Wallington – all NT
Other Attractions
Brinkburn Priory, Coquet Island, Coquet Valley, Corbridge Roman Town, Farne Islands, Grace Darling museum, Hadrian's Wall - Chesters, Housesteads and Vindolanda, Ford and Etal, Hauxley nature reserve, Heatherslaw Light Railway, Holy Island, Howick gardens, Ingram Valley, Kielder, Warkworth Hermitage, Woodhorn museum
Beaches
Alnmouth, Bamburgh, Beadnell, Boulmer, Cocklawburn, Cresswell, Druridge Bay, Embleton, Holy Island, Low Newton, Longhoughton, Newbiggin, Seahouses, Seaton Sluice, Warkworth,
Children
Northumberland Country Zoo, Northumberlandia, Whitehouse farm centre
Places to eat
Ephesus, Morpeth
Nadon Thai, Morpeth
Tandoor Mahal, Morpeth
Lal Khazana, Shilbottle
Cook and Barker Inn, Newton on the Moor
The Running Fox, Felton and Longframlingtom
The Village Inn, Longframlimgtom
The Lord Crewe Arms, Blanchland
I wouldn't bother going to the Metrocentre. It isn't that straightforward to get to without a car as it isn't near a railway station or metro station.
If you want to go shopping I would drive to Kingston Park and get the metro into Newcastle and get off at Haymarket or Monument. Eldon Square is an excellent shopping mall in the middle of Newcastle, and IMO better than the Metrocentre which has gone downhill a lot since covid.
Also, look up tide timetables for the week for Holy Island so you can plan the right day to go there. Northumberland has an abundance of National Trust properties, and if you like visiting NT properties joining would pay for itself while you are there. IMO the best NT place to visit is Cragside. I think it is the jewel in Northumberland's crown.