Having lived in Yorks for 43 years, I am pretty familiar with the county , and I cant fault any of the suggestions.
York is a must similar to Chester for those who have never visited. Whitby is stunning, If you like to walk I strongly suggest the walk from Robin Hoods bay to Whitby part of the Cleveland Way ( I think ) Amazing views around everyturn and when you enter Whitby you will pass a smoking house for traditional oak smoked kippers...... Europe are try ing to ban it ( but another thread).
Jet a semi precious stone is unique to this are, and many jewellers offer their ware in the town, Southside, the cliffs are full of ammonites if you fancy a bit of fossil discovery.
One trip well worthy is to leave Whitby to the North, take in Sandsend and if the Norths Sea is agitated the rollers are fantastic and you will often see surfers , park up kettle on and enjoy ....A little further North is Staithes ( similar to Clovelly in Devon I think ) a hidden jewel in Yorks and home to The Staithes " School" of artists......cobble fishing boats unique to the area lilt at low tide and often you will find artists painting, Capt Cook apparently also served his apprenticeship here before he left Whitby and discovered Australia>
Slightly inland is Goathland the village where Heartbeat was filmed ( gives you an idea of the scenery )
As others correctly suggest West Yorks, North Yorks etc have plenty to offer too....Gods own county it is often called ......Better than that .........If Carlsberg made counties .....Yorkshire would be the blue print
Channa