You may want to do some research on your own since your question is generic and only you know what you like to see and do. I normally start with TripAdvisor and look at “things to do” to get an idea of what is in that area. I bookmark places on Google maps and by doing that, there may be places on maps I overlooked that are of interest. Once you’ve done that you can come back with more focused questions. How you’re getting around could determine how easy it will be to get to the places you want to visit.
In New Castle Upon Tyne there is a castle and old mill and just outside of the city is the Angel of the North. A little further out is Beamish, the Living Museum of the North. Durham has a castle and cathedral. Believe the UK TV series Vera is filmed in Newcastle Upon Tyne area. I haven’t been to any of the places I mentioned, but they are on my “to see” list.
A few miles north of Bamburgh castle is Holy Island. It’s nice to visit. We visited it last September. How long you spend there depends on whether you walk over to the castle or not. Ensure you check the tide tables so you’re not caught on the island when the tide comes in. There isn’t a lot to do if you miss getting off the island and have to wait until the next low tide.