When I lived in Sheffield I often seemed to end up looking after friends' kids for the day. (I didn't really mind - it gave me an excuse to be a big kid for the day too!). York was one of the places I visited many times with youngsters.
As has been suggested, the National Railway Museum is unmissable (and, as it's actually an outpost of the Science Museum in London, it's free!):
https://www.railwaymuseum.org.uk/
(Allow at least a couple of hours for your visit).
The Jorvik Viking Centre is pricey but worth a visit:
https://www.jorvikvikingcentre.co.uk
(Consider booking online to avoid the queues)
Clifford's Tower is (at typical English Heritage prices) possibly rather expensive for what you get for your money (which is basically the chance to climb some steep steps, look at the last remaining part of York Castle and enjoy the views) but I'd still include it my list of recommendations if the weather's good.
The neighbouring Castle Museum can appear pricey (at £12 per adult) but kids get in for free, so it doesn't workout too badly for families. The most popular parts are those which are laid out as streets from the past:
https://www.yorkcastlemuseum.org.uk/
Obviously, simply walking around the city walls, and along The Shambles, should be included in your visit. There are also the open-top bus tours to consider or, much better in my opinion, sightseeing river cruises:
https://www.citycruisesyork.com/
I'm not sure what you're going to do with the dog though; I've never thought of York as a particularly dog-friendly city.