As a Yorkshirewoman (Scottish Mum) I am constantly amazed by the variety of my much loved county. Most of the Winter Walks are in N. Yorkshire - but that in itself is huge. I could drive for almost 3 hours before running out of Yorkshire to the West.
You might think of centring somewhere in the Vale of York where you could reach East or West, Thirsk perhaps (World of James Herriot). This would also give you access to the Great Yorkshire Forest (Dalby) near Pickering, where there are tree walks and bike trails for younger members. It really does depend on what you want.
Tonight's walk was not far West of Skipton - another good centre. Should you head a bit further North you have Settle and Ingleton and the 3 Peaks area - wonderful walking and the 'Falls walk' at Ingleton would be OK for an 11 yr. old. Then there is Malham Cove (heaves with tourists) or up towards Hawes and the creamery as has been said. So very much - and I haven't touched the coast!
Whitby, of course, but plenty of other places. If you are into birds there is Flamborough Head.
I'm very willing to give advice and help, but you need to narrow your search a bit! We (E. Yorkshire)took a holiday in the Forest of Bowland area the other year. :)