Depends a bit which direction you are from York. If it's the coast side, then Whitby, Sandsend, Filey, Flamborough Head...
The moors, Goathland, steam train, all that lot.
Near York there's Castle Howard, the arboretum next door, farming museum, Yorkshire Lavender, Breezy Knees gardens (amazing, and massive), Beningbrough Hall, Newby Hall...
If to the west, various abbeys – Fountains Abbey, Jervaulx Abbey, Riveax – Knaresborough, all of the Dales, er... there's just loads!
We moved to York six years ago and haven't scratched the surface of the things to do.