The Castle Museum with its Victorian street is very good, but not free. Gift aid the entrance fee and the ticket is valid for a year.
National Railway museum is free.
Dungeons is, in my opinion, over-rated and expensive. If you're desperate to go there look out for 2 for 1 tickets. They're all over the place.
The art gallery, too, is free with some interesting paintings and ceramics collection.
Yorkshire Museum is interesting in parts and not free.
Yorvic centre is OK if you;re interested in the ancient history of York. Can be long queues and fairly expensive.
There are two good antique centres on Stonegate if that's your scene. Free entry, obviously.
Whatever you do you'll enjoy yourselves.