Building work is now finished and the little train to the sealife is lovely. The park opposite has peddle dragons for the lake and there will be an old style band on for the weekend usually. It will be packed and parking will break your heart. Go early.
Try get a half price voucher or something for sealife - they are expensive.
Look at the museums, the Rotunda does fossil walks and info tours on how to look for fossils.
PlayDale farm - very good, lots of indoor play if weather pants.
Betton Farm - nice little play area and lots of info on bees and honey etc. Nice grub!
Other things:
Pickering - most visited steam railway in UK. It is lovely. Just chug up to Goathland, have and ice cream and come back. Or you can do the longer jaunt to Whitby. Or just hang about and soak up the atmosphere. There is a website you can check out.
For lunch a farm cafe called Cedar Barn a mile outside Pickering. Has outdoor play, fruit picking if the season is right an the food is just amazing. Family friendly - you can ask for pretty much anything and they will see what they can do. Bookings only Sunday.
Thornton-le-Dale 3 miles from Pickering and cute as a button. Ducks to feed and icecreams, a pot decorating place some gift shops and a little wander. That's about it.
Dalby Forest - just brill but maybe a bit old for your dc. Great for a bike ride, also the best adventure playground for miles. Costs a £7 in to the forest. Excellent visitor centre and bike hire.
Castle Howard - my all time fave. Fab fab fab adventure playground, amazing grounds, tractor type drive down to lake, lovely gift shops, amazing house - my idea of utopia.
Tis a magic place for preschoolers around these parts!
Robin Hoods Bay - ace for pretending to be pirates and smugglers and soaking up the atmosphere