If you get bored, the 843 bus from opposite the train station costs £6.50 return for adults and takes 55mins to get to Leeds.
The Millenium Bridge is built to look like a bicycle wheel, it’s a walk down the river from town, then cross down the other bank and visit Rowntrees Park for a relaxing walk (pond, small kids and medium kids play areas)
York Designer Centre (indoor shopping centre with covered play area outside for small to medium sized kids) is just past Fulford. Buses run there, or if you’ve brought your car they have a massive car park.
Strensall Common is a few miles out of York if you drive there for a family walk/picnic. Can’t even remember if the army are still there (if they are, stay off the firing ranges) If the weather is bad (read good) then bring bug juice for the mozzies and watch out for the occasional grass snake (one bit my hand 🙁 )
You can rent little motorboats on the river for extortionate prices. The Castle Museum is fine for residents (who get in free and aren’t ripped off) The Minster costs a fortune too, just attend a service to get in for free (don’t wander round mid service though!)
Best asian place to eat in the centre of town: Happy Valley. Next is Evil Eye (head chef is Indonesian and her food is rather good)
The Jorvik Centre is a lot better nowadays. It’s worth it I reckon.