Well depending on how long you're going to be here...
York Minster is a must. In York itself, I would say that Jorvik and the National Railway Museum might go over well with children -- though I've never been to either.
Leeds is very well known for its city centre - fab shopping area. And the Royal Armouries museum (also possibly Thackray Museum) would be a good bet for the kiddies. A link with other museums in the Leeds/Bradford area:
http://www.10best.com/Leeds/Sights_&_Activities/Museums/index.htmlLiving here, some of my favorite things in Yorkshire have been:
Walking in the Yorkshire Dales and North York Moors
Harrogate & surrounds -- I particularly loved Fountains Abbey, which is a World Heritage Site:
http://www.fountainsabbey.org.uk/Harrogate is a Victorian spa town & there's a Turkish baths/spa there. (Ilkley is also a Victorian spa town.) And I love walking around the Nidderdale and Pately Bridge area.
Also I've heard good things about Knaresborough Castle and Ripley Castle, and Ripon (Ripon Cathedral is there) where "setting the watch" occurs nightly at 9:00 p.m. with the blowing of a horn in Ripon Market Place.
Further afield (from me) -- I loved the North Yorkshire coastal towns, particularly Whitby (where Dracula came ashore into England!

) and Scarborough. There's also The Deep (aquarium) in Hull, though I'm not sure that Hull is technically in Yorkshire any longer -- could be wrong...it might be East Ridings, I don't know - I get confused about all that.
Here's the Yorkshire tourism link:
http://www.yorkshirevisitor.com/ (Good link, there's even a bit with suggested itineraries.)
Let me know if you have any questions & I'll try to help out or find an answer (ask someone, etc). I love the countryside around here & places like York, Harrogate, Leeds, etc.

(It's James Herriot country here too!)