Got to be Bamburgh and Lindisfarne for a taste of everything. History, beach etc; they were to p of every visitors list when I lived in the area.
Take them to Roker beach at low tide, and catch loads of crabs, then go back up the bank, and place said crabs in the middle of the road, then you and the kids can guess how many will make it to safety, the kids will lovw it, honest
So many choices. Durham Cathedral, Alnwick, Cragside, Bamburgh, Whitley Bay, Barnard Castle, High Force, Roseberry Topping, Beamish to name a few. Shame Lambton Lion Park isn't there any more though.
Join the national trust, I'm a member and the kids for the three of us it costs me around £50 for a year. That is free entry to any national trust site all year round. They have become quite kid friendly too - perfect for 4-8 year olds. I would recommend - Gibbside, loads for the kids, good play grounds, nice walks good coffee shops they even do beer festivals. Wallington- again loads to do, plenty for the kids, good play areas, good walks. Cragside is around the best, you could easily spend a couple of days there. Souter light house, great on a sunny day. They also have the roman fort on hadrians wall. housteads which is great for exploring. Seaton Deleval Hall is up and coming only half a day needed there but you are close to the seaside as well. Fincale abbey is great as already mentioned but there is a new place opened right beside it, adventure Valley - lots and lots for your kids there, great play grounds, mini pettnig zoo, horse riding, tractor rides, a great big tyre slide thing which is great. lots of fun. Near to Beamish is Beamish wild (look for offers on groupon) decent enough day out. A little further out moving to North Yorkshire, The forbidden Garden is a great day out really quirky but you have to book in advance, Fountains Abbey is brillaint, Brimham Rocks is one of the best places you'll go. Castle Eden Dene is good for walking.