Alnmouth has featured on Vera and would be a good base. It has a lovely beach and if you can't get a boat trip because of the weather it is handy for driving over the crossway to Holy Island, which you can see from the beach. It also has a train station, which may be useful, particularly if you are planning to go into Newcastle.
Alnwick castle (a couple of miles from Alnmouth) is a good day out, as you can go round the castle and the gardens on one (pricey) ticket and between them they would easily fill a day. During school holidays it does tend to go full-on HP though, so you might want to check first. Warkworth Castle is a ruin, and is lovely. Warkworth Village has lovely cafes which would cater for vegans. Check if there are still trips over the river to the Hermitage, which is also worth a visit. There used to be a man with a boat who took you across, and you can see where Harry Percy was christened and a cave that was lived in by a hermit for year.
It's EH, so free if you are a member
If you go into Newcastle, definitely go to the Great North Museum (on the outskirts of the city centre next to the University, and also free). Again, you can spend ages in there - it's really good, and you're in the city for food and possible shopping afterwards. You can get a Metro from Central Station to Haymarket and it's a five minute walk from there. There are trips up and down the Tyne too - one industrial one looking at disused shipyards, which might not be your thing

, and another going the other way to spot otters and things. Tourist Information will give you a timetable. If you want to do that, the trips leave from Quayside, and you could leave the Museum, cross the road to Haymarket bus station and get a Quaylink bus from the first stand you come to. That will put you off on Quayside, which has access to the Baltic Art Gallery, or you could cross the Millennium Bridge and go to the Sage (a music venue). Both free entry, although there may be a charge for some exhibitions in the Baltic.
If none of that appeals, or if you have more specific questions, just shout.
ETA Cross posted with Chicky, but agree with her suggestions too. Apart from the Farne Islands, which I loathe, as they stink and you are guaranteed to get birdpoo on your head.