Castle Howard (20 mins) is pricey but has an excellent adventure playground which would be particularly good for your older child.
Scarborough (35 mins) has a Sealife Centre, a castle, great beaches - North Bay is the best for little ones, boat rides, arcades etc so that is a good day out. Alpamere water park is reopening at the end of July, it's pricey but has a wave pool, water play area, outdoor heated pools etc.
The naval battle at Peasholme Park is good, it's a reinactment if a WWII NB aval battle with pyrotechnics and model boats, normally on certain days during the summer. There's a little steam train that runs from the park across the road to the Sealife Centre to save little legs too.
Robin Hood's Bay (45 mins) is amazing. Beautiful village with a great beach. Park at Boogle Hole, stop off at the Youth Hostel for cake and walk the mile to Robin Hood's Bay. Check the tide times as this will be a lot easier if you walk along the beach rather than up the hill and along the cliffs. watch for the tide as it comes in quicker than you'd think.
Whitby (40 mins over the moors) has a great beach again, boat rides, and there is a nice playground outside the small museum at Pannet Park.
Obviously go into York one day (around 40 mins if the traffic is good, park in Foss Bank car park). Walk along the walks, Jorvik possibly, Roberto's for ice cream, hire a boat, lots of nice family friendly restaurants.
Dalby forest is very near Pickering.(15 mins) You have to pay to get in but there are lots of walks, a child's trail, an excellent playground, a stream to paddle in and a Go Ape ( maybe more for your elder one!) Stop off in Thornton Dale afterwards for an ice cream at Balderstones.
Pickering has a small castle and a stream train as well as lots of nice cafes and ice cream shops.
Helmsley (15 mins away) has a ruined castle that's good for a run around, ice creams etc Also Riveaulx Abbey is a couple of miles away and although ruined is beautiful and a good place to let off steam.
North Yorkshire Water Park at Wykeham (30 mins) has an inflatable water park with a toddler section. They have electric boats too.
There's a soft play at Ruswarp near Whitby if you get rain. It's called Mini Monsters and is great. You can also hire row boats across the road
Hope this is helpful! There's lots to do, just mist if it involves a bit of a drive. Hope you have a great time!