Hi rebklein -
I suppose that it all boils down to what your personal preferences are. Let’s face it, there is enough in York to occupy you for three days easily. With kids I would definitely recommend The Railway Museum, currently free to enter (donation requested but ultimately that’s up to you) as they have a chance to get up close to giant steam (and diesel/electric) engines and peer onto the footplate of the likes of Mallard, still the world record speed holder for a steam locomotive. There’s also several platforms of various incarnations of ‘Royal Train’ rolling stock so you can see how the monarchy travels!
With the Dales, there’s so much of it that it’s difficult to know where to start. But my personal preference would be Malham as there’s a lot to see and do in a relatively small area. It’s not too long a drive from York to begin with. And the car parking is (semi) reasonable price wise, and there’s a visitor centre which explains the area and where you can buy maps and guides etc., (amongst other things!). I would avoid it on a weekend as it gets really, really busy, but most week days there’ll not be as many people about. Richard mentions the ‘Malham Round’ which leaves the village for Janet’s Foss through fields and woodland and emerges near Gordale Scar (mobile refreshment van usually in attendance!) and either climbs the waterfall through the scar (maybe not for young kids but investigation definitely required) or by passes it on a green path running behind the refreshment van and crosses over to the limestone pavement atop Malham Cove (care needed crossing this especially if wet). From here steps lead to the foot of the Cove and back to Malham or you can head for Malham Tarn via Watlowes dry valley - there are various ways back if you don’t want to retrace your steps. Plenty of accommodation in the village.
However, if you haven’t booked accommodation yet you might be restricted to what you can find, so it might be sensible to let your itinerary be guided by what you can book. York is very popular, especially during the summer and can be heaving with hen/stag do’s at weekends particularly and the same goes for when the horse racing meetings are on. The good news is the bits of York you’ll want to investigate are not on the agenda of those I’ve just mentioned.
I always recommend Fountains Abbey - it’s expensive to enter but will occupy the best part of a day, and is probably worth it - I believe there’s a ‘family ticket’ to ease the burden! The Forbidden Corner is a great fun - I’m approaching sixty from the wrong direction and I thought it was a hoot!
If York is difficult to book, consider Harrogate or maybe even Grassington (I’m told there is a super air bnb with hot tub just off the square!) which is on the edge of the Dales, or even Ripon or Masham. All are within striking distance of the Dales and are relatively short drives from York if you can only day trip.
Let us know how you get on and what you decide. Where you can book will dictate where you ultimately go. Feel free to ask away and if you want to PM me for info, don’t hesitate.
Hope it all comes together for you!
Ian