I'd stay in/near Filey but as it's only about 7-10 miles from Brid then you can 'do' both and there's Scarborough about the same distance in the opposite direction to Brid.
We like the village of Hunmanby and there are quite afew nice sites around and about.
Centenary Way at Filey is supposed to be a great site and always gets good reviews and comments. It's just about as near to the beach as you can get too.
No idea on the CL but Saxdale House Farm is a nice quiet little site at Hunmanby. Not as cheap as a CL but I think about £10 a night £12 with electric (think we paid £12 for tent and family of 4)
We just back from crows nest at filey that was good, after some reviews should some bad bits we found it brill... £20 a night and entertainment for kids and adults was good... Centenary way ok too
Then you are in between Scarborough and bridlington
personally i prefer Brid to Filey but its all a matter of personal taste. More to do in Brid but Filey is a lot quieter unless you are on either of the Haven sites, Primmy Valley or Reighton Sands, then it is anything but quiet!!
Don't forget to leave a review of the campsites you have visited this year or last!
Personally I prefer Filey because its quieter (dont do Haven type sites) but Brid is nice for a visit and much improved in recent years.
If you do find a good CL there Id love to know Mrs B. I have earmarked one further up the coast at Ravenscar, but Ive not stayed there yet so cant recommend it.