Oh now how long have you got? This is home territory for me. There are many places which offer different things. Personally I love Saltburn, which is further up coast. There you have a good beach - see surfers having a go - a pier and a funicular to take you down to the beach from high up. I have a disabled badge and find it best to park up the top, with views out to sea with my binoculars etc., It is a quieter place with a whole load of different cafes , restaurants, a local market, a railway station etc. Anywhere along the coast it is worth having your bus pass as you can have lovely trips on double deckers with a grand view over the sea. Bridlington has a harbour and I think still has the good fresh fish shop, which is owned by a family who fish with their boat and so you definitely get a very fresh fish. They sometimes have lobster tails at a reasonable price and I , being yorkshire of course, prefer to bring my picnic box with me and then can get the lobster tail, go up on the moor and enjoy a picnic up there.
Whilst over that way you can actually visit Warter , if you are a David Hockney fan and see where he painted them. For something quite different you can go across to Hunmanby on the coast and look for Reighton Nurseries, a must for any mad keen gardener . Now due to health reasons havent been recently so you might need to check it out. But I usually go with a sandwich and drink in the car, everything unnecessary taken from the boot to allow me to spent up to a whole day, going round the nursery. You will find plants with weeds in the pots but oh my the prices are good, and I have always quarantined any plant I bring into my garden, and the good prices mean that I am like a child in a sweety shop.
Now if you should come over to North Yorkshire at either the last week in May first week in June, the wonderful Swaledale festival is on and you can have a marvellous time going to concerts of all types and all sorts of walks and poetry days etc etc. That is all up Swaledale andWenslydale, and is held then when the wonderful hay meadows are full of wonderful flowers . I used to live in Richmond and it is easy to go up and down the dales , or go across via probably Stoklesy , which is also a good place to look round, and then across to which ever part of the coast appeals , so you might go to Redcar races or as I say spend a day at Saltburn with lovely beaches or park your car sensibly and enjoy a lovely ride down to Whitby or Scarborough on the bus, where you will have great views, no worry about parking, and it is a great trip.
If you were a fan of Heartbeat, you can go on the train or by car to Gothland etc . Or if you come at this time of year, then I would recommend coming to swaledale for the first wednesday in september , when the great Muker show will be on. I admit to being a little biased, but it is a truly lovely small local agricultural show, but where you will have the great Muker silver band playing all day, the fancy dress, the tents with the flower and vegetables and craft shows, then watch the sheepdogs coping with the very stroppy Swaledale sheep, (which I think all sheepdog trials should have to work with) then see the dales ponies etc etc. Sit with your binoculars watching the fell race running above you and adults and children of all ages competing. Then finally the annoucement will be made "the show is now closing, so make your way down to the Farmers Arms folks" the Muker silver band go down, we dish out the hymn books and you can go and get a drink from the pub and come out to join with us all and we never let a bit of rain put us off, as we sing many old favourites and of course most importantly "Swaledale, Beautiful Swaledale" There are plenty of other shows on around that time but as I say why go elsewhere when you can have the best? Then there will be the various cycle races from the over the buttertubs, and the Lloyds race and all sorts of things to watch . You will be able to go to see from some of the ports, and of course the simplest way to go to York from whereever you are staying is to either, drive to the park and ride nearest to your place and get the bus into York, or just go on the bus from further out. Of course we have various Bettys in different towns but you can go to the original one in Harrogate and can visit RHS Harlow Carr , where you will see some lovely plants and see what will cope in a colder climate etc. Oh dear I sound like the TI and havent crammed in half of what there is. I hope there is something to appeal to you, I dont go a lot to Scarborough but did love Peasholm park and my uncle used to race on motor bikes on Olivers Mount.
I hope that something catches your eye and you venture over here . Who knows if one or two people come we might have a GN meet up for coffee in Guisborough or Northallerton or for those who enjoyed the James Herriot books and films there is Thirsk where you can visit all things related to that too. Oh lucky you if you are coming for the first time!! Enjoy yourselves
We moved away but came back 6 years ago after 40+ years. Sadly the place isn't what it once was but then neither is our home town or the one we left to come back but thankfully for businesses the great weather has been wonderful for visitors 
I also went to Plymouth on what would have been Thanksgiving in the States as I read in the paper that some restaurants did American style Thanksgiving dinners. Apparently a lot of US visitors at that time (where the Pilgrims set off from England). Then in 2003, I was teaching nursery nurses in Kent and we showed some videos from TV that were relevant to the course. One show that I expecially enjoyed was set in Scarborough which I thought was so lovely. A few years later, when I had gotten hooked on Heartbeat, I took two friends and we did a tour of Goathland, Thirsk (Dr. Herriot), and Scarborough. It was wonderful and I really enjoyed the North. Both friends have sadly died but I have lovely memories and some photos.