It's such a big area. West and South Yorkshire have their own district feel away from North Yorkshire. I'd maybe focus on York and the North Yorkshire coast. Maybe a couple of nights in York then over to Whitby.
York has loads for families and great evening choices. Yorvik is fun for younger kids. But pricey
Depending on what your family enjoy and what's open fully in February, Id definitely recommend some of the wonderful houses,castles and abbeys of the area. Castle Howard is so iconic. Early spring is stunning at Fountains abbey with the daffodils out.
The smaller market towns and cathedral cities are special. Several to chose from. Ripon, a lovely little city that's quieter than York, Thirsk where the original James Herriot had his vets practice, Richmond and the villages of Swaledale and Wensleydale. Also the waterfalls of the Tees Valley will be in full force
Head over the Moors to Whitby. It's Heartbeat land here and there's a couple of lovely steam railway options as well as cute villages . Staithes is glorious but much more to do in Whitby. Dracula and the Goths, Whaling, fishing fleets. And the wonderful abbey. Plus fish an chips at the Magpie. Book ahead if you don't want to queue.
If you're feelin fit the Cleveland way passes from Whitby by the abbey along the cliffs to beautiful Robin Hoods Bay. It's about seven miles and it can be a bit hairy if the winds up. Careful with kids and dogs as it's very close to the cliff edges. Alternatively the cinder trail is the old railway line path that is easier and more sheltered. Nice pubs for lunch in Robin Hoods Bay and a easy bus back to Whitby.
I grew up nearby and spent every summer in Staithes so happy to help. Happy holidays.