Take a steam train to Pickering and back. You might even be able to get afternoon tea in a Pullman.
Walk the length of the beach to Sandsend (several sweet cafes - the best is Wits End, tucked next to the car park toilets!) and back.
if you’ve got a car, drive over the moors to Egton Bridge. Decent pub grub, beautiful church and stepping stones to jump across.
The Abbey is a must and the adjoining museum is good on what life was like for monks. The 199 steps up must be done, and counted.
Pannet Park museum has some really weird and wonderful exhibits, including almost complete dinosaur skeleton.
Lots of quirky shops to look round - you can lose half a day easily.
Boat trip - might see seals.
The Magpie is great, but if the queues are too horrendous, Trenchers near the station matches it for quality of F&C.