The National Railway Museum is in York and is free of charge if your into that, then there is the Jorvik Viking Centre that is quite interesting that recreates how the Vikings used to live when they inhabited York.
There is also the York Dungeons that re-create the various torture methods used in the past.
Also York Minster (Big Cathedral) is free too, and there is a Castle to look round or you can go for a walk on city walls.
Not far from York is the Yorkshire Air Museum at Elvington. Has a fully airworthy Victor Bomber too, that isn't allowed to fly, but does do some high speed taxi runs up and down the runway from time to time.
There are also plenty of shops as well.
You should also be able to find some cheap hostel to stay in York as it's quite popular on the backpacker trail. Lots of good bars/pubs too.
"She Rolls, 45 knots, 90, 135, nose comes up to 20 degrees, she's airborne - She flies, Concorde Flies"