Whitby itself is a pretty cool place to photograph. It's a harbour and the sunsets can be pretty amazing. Although it's on the east coast, the coast faces almost north. As the sun sets, the remaining light illuminates the hill where the church and abbey ruin are. I have not had much chance myself but I suspect there are some good photos to be had around that time.
My family and I usually go to Bridlington but for days trips, Robin Hoods bay (south of whitby) is good when the tide's out (great for rock pools with reflections). Bempton cliffs is an RSPB site which is usually home to a LOT of nesting birds. Possibly about the right time of year to see puffins if birding is your thing.
I've not been to those places just to photograph as I'm usually with family but the whole coastline is quite picturesque. My last trip, I had a go at long exposures of the stars which came out pretty cool (for a first attempt)
Hope that helps, enjoy your trip