Sorry that it went a bit quiet. I am in the middle of GCSE marking so have had to take a step back from MN in order to concentrate (clearly hasn't worked today)!
Have you found any good local groups? Have you looked on the listings here, although they aren't exhaustive yet.
Failing that the York council website, and particularly the Family Information Service, is really useful:
Council
The Minster used to give out Residents cards with proof of address, but I think that that's changed now. Go in and ask someone, they're really friendly in there and I was allowed in a few times before I got one.
Library cards are pretty essential. They'll get you into The Yorkshire Museum in Museum gardens and The Castle Museum near Clifford's Tower for free. Definitely worthwhile. You have to pay a small amount for them though so take some change with you when you get them.
There are so many great things on in York for families with young children. Sadly the library in town is closed for 6 months [shudder] for refurbishments and the WONDERFUL gallery will be opening again in Jan 2015 after a 2 year refurbishment programme. My kids practically grew up in that gallery.
I'm sure that you've found all of this out already, so apologies again for being a bit rubbish.
Oh, my top tip would be to take the kids to the Theatre Royal café. There's a children's play area there which is beautiful so you can just switch off with a coffee while they play. Essential!