Cycle York itself, including along the riverbank. V pleasant indeed....York is very, very cycle-friendly.
Here's the cycle route map for the city:
and their 'leisure route' cycle maps:
and how to cycle the Solar System (I've done a bit of this on foot and it's good fun!):
Here's a link to N Yorkshire cycle routes:
If you're keen, maybe cycle out to Castle Howard (about 15 miles)?
Also, minus bike:
Get the bus to Malton and visit the excellent Eden Camp:
Bus info: http://www.yorkshirecoastliner.co.uk/
Take the train to Scarborough or Whitby, or to Pickering where you can pick up the North York Moors railway:
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