York - Jarviksholm, the Walls, the Shambles is lovely for shopping. A visit to Betty's Tea Room - unless you're going to Harrogate, which us where the original Betty's is! I recommend the kedgeree for brunch 😊
Leeds - The Armories (part of Imperial War Museums) is excellent, very informative and entertaining. Some days they have outdoor exhibitions such as jousting and birds of prey. Indoors, last time I went they had hourly live demonstrations of samurai fighting.
The river walk in Leeds is very pleasant.
Sheffield - best place in Yorkshire, but I'm biased 😊 The city centre around the Town Hall and the Peace Gardens is beautiful, kids usually run around through the fountains so take some spare clothes! You've got the Winter Gardens as well, and the entrance to the Millennium gallery is within them. The cathedral is wonderful. There are a lot of large parks - Rother Valley country park does some watersports stuff including that waterski thing where they're pulled by a cable rather than a boat. (Cable skiing?)
I'd also recommend getting over to the Peak District and visiting some of the cave systems. Speedwell Cavern is really interesting - it's about 200 steps below ground and now flooded, so you do the trip in a row boat wit the guide. It's very atmospheric, can be quite eerie and somewhat emotional hearing the stories of the children who were sent into the mines and often died there.
Kids otoh often favour the cave known as "The Devils Arse" - because when the wind blows from the wrong direction, it makes a satanic farting sound 😂
Yorkshire Wildlife Park is to the east of Sheffield and also a good day out. Not a huge place but you do get up close and personal with a few of the animals.
And in Rotherham you've got Magna - a science museum with a focus on steel working.
Have fun!