May is a wionderful time to visit and the hay meadowswill be covered with wild flowers.
You are literally spoilt for choice here... Of the top of my head, places like Skipton, Settle, Grassington, Hawes, Reeth, Richmond... even Barnard Castle, which is actually County Durham rather than North Yorkshire, but is still a good base for Teesdale and Swaledale. Wensleydale is just over an hour in the car.
Each place is very different with different attractions and scenery. You can't go wrong, where ever you decide to stop. (They will make the Cotswolds seem tame!!)
If you like museums I would possibly head to
Barnard Castle and the wonderful
Bowes Museum. (Don't miss the wonderful
mechanical silver swan.)The town itself is a thrivng local market centre with a good range of shops and an
attractive place to walk round. It has a ruined castle and ruined abbey close by. It is also convenient to visit Durham as well as Teesdale and places like High Force... Richmond with its ruined castle as well as Easby Abbey and St Agatha's church with its medieval wall paintings, and Swaledale are easily reached from here too.
Auckland Castle, once the home of the Bishops of Durham, is only a short drive.
Raby Castle is just a few minutes drive and is still very much a much loved family home with a very long and illustrious history (Go and see the family tombs in Staindrop church.) The inside is magnificent and it has lovely gardens to walk round and a very good tea troom.
Eggleston Hall Gardens are delightful and completely off the tourist radar.
This area tends to be ignored by visitors who head to the southern part of the Yorkshire Dales, but it does have a lot to attract it. There is also plenty of good walking too.