There's Eden Camp near Malton, which is a museum about world war 2. About 40 mins from Scarborough. Just down the road from there is Flamingo Land, amusement park and zoo.
Espresso Yourself is a nice cafe.
Agree about Flamborough Head, about 30 mins from Scarborough, lots of sea birds to see, although it's better in the summer when the puffins are there. There's also a lighthouse.
Sewerby Hall has nice gardens and a tea room, also a zoo, but might be closed in winter. There's Bridlington animal and bird park, again might be closed. Burton Agnes hall near Bridlington. Castle Howard on the A64 on the way to Scarborough.
Lots of amusement arcades on the sea front in Scarborough. A small funfair but again probably closed in winter.
Go to the Harbour Bar on the seafront for lovely ice cream and hot drinks, they do amazing knickerbocker glories.