Scarborough North Bay is really lovely and there is a great surf bar called Blue Crush just opened there recently. (south bay is typical candy floss/chips/amusements). great place called playdale Farm at Cayton a couple of miles south of Scarborough.
Flamingoland, near Pickering really worth a visit. Loads to to do and see,..even for pre-schoolers.
Nice to go on the North York Moors Railway - steam trains and some beautiful little stations (levisham, goathland etc) with great walking. You can get a train Pickering through to Whitby (also pretty, and home of the Endeavour)
Ravenscar to Robin Hoods bay is magic - as long as there is no sea fret in.
Filey is really pretty too.
Agree with Vicarinatutu about Dalby Forest (I live there!) great for mountain biking, go see the Bridestones, lovely little tea room called Jingleby Farm at the top end near the lake...there is a go Ape and good visitor centre
Loads of places to stay to suit all budgets - carvans, lodges, reasonable B & B, cottages, hotels
I am not a Yorkshire lass but I could go on all day about the part of north Yorkshire you are talking about.
Come on over...its magic!!