Whitby is a fab area. I can't go there without doing a bit of fossil hunting on the beach
The trick is to crack the pebbles apart, you're sure to find one. It'll bring out the big kid in you.
The Magpie is a must. About £10 for fish and chips but well worth it. Mmm
And if you fancy venturing up the coast a bit Sandsend, Staithes and Runswick Bay are all lovely little villages.
Hope you have a good time.