I live in Whitby :)
I have 3 children 6 and under.
Things I would recommend...
Sandsend beach- best beach for children and you don't have to walk up the cliff at the end! Although it gets VERY busy if the weather is nice and there's not that much parking.
North Yorkshire Moors railway @ Grosmont, my children just like to go and watch the trains from the platform and visit the engine sheds, we rarely actually go on the train.
Sealife centre in Scarborough - although it's VERY expensive (you can usually get deals on tickets though, especially if you buy a local paper)
Falling Foss - waterfall, walk and a lovely tea garden which is very resonable.
Danby Moors Centre - FREE, outside play, gardens, woodland trail and a Moors museum.
Mini Monsters soft play, expensive but good.
Robin Hoods Bay is lovely. Runswick Bay also lovely but no quite as busy.
Whitby Abbey.
Dalby Forest
Castle Howard
We're local and like the Magpie, although we tend to go off season to avoid the queues, they also only take a limited amount of bookings. Most of the fish and chips shops are good, try Fusco's as an alternative.
There's absolutely loads to do round here with children, hope you have a good time :)