Yorkshire

I believe many of the scenic Yorkshire areas are in North Yorkshire, which isn't always easily accessible by public transport. Nonetheless, Harrogate (inland) Scarborough and Whitby (both coastal) are popular destinations in North Yorkshire. Perhaps you could take the train to York or Leeds, and rent a car there to drive through the Dales?

We enjoyed Richmond and a drive through Swaledale. Whitby is often recommended but I haven't been there. York itself is a great destination, worth at least a couple of days if you have the time. There's a train through the moors between Whitby and a town north of York that I can't remember now -- not a regular "transit" train but a scenic ride.

I am heading back to Yorkshire this July for a few days stopping in York to visit the railway museum, shambles, cathedral, then taking the scenic North Yorkshire railway which stops at some pretty towns Goathland (Harry potter station in first movie) There is Robin Hood’s Bay, a small fishing village and a bay located within the North York Moors National Park, five miles south of Whitby I am also going to visit the Yorkshire Dales and staying near Settle. I was born in England and go across the pond every year to visit family around London. Yorkshire is one of my favorite destinations in England. You would certainly need a car to get around the Dales or North Yorkshire Moors. But it is beautiful and well worth it. B&B would be the way to go. The dales have small roads and stone walls (keeping in sheep), so driving is a little tricky, but as long as you drive slowly, you will be fine.

We love to stay in Grassington in the Yorkshire Dales National Park. It is a lovely little village. We did not rent a car. It was/is accessible from Leeds and Skipton by train/bus. We have also just arranged for a taxi, most of the time, when arriving and leaving. The area is gorgeous! There are lovely vistas and walks through the hills and hamlets. The village is quaint. We stay in a 17th century miner's cottage. We do fine catching the bus back or to our hikes, if we need to do so, but again, have been known to call for the Grassington taxi driver to pick us up where ever we are. I highly recommend a few days in this beautiful village.

Edit: And I will add, in agreement, that my British son in law's favorite place in the UK is Robin Hood’s Bay, where they are headed next week, again. His mother loves Whitby. I have yet to go to either place.

Yorkshire Dales National Park:
http://www.yorkshiredales.org.uk/

Keld is an interesting small village in the northwestern part of the Dales. Limited overnight accommodations and no pub. Apparently a temperance advocate some time in the 1950s (?) bought the village pub and shut it down. I understand that the Keld hostel has been purchased since our visit and reopened with lodging, restaurant and bar. Keld is the headwaters of the Swaledale which runs dark because of the peat. Lead miners in the 1800s use to leave Keld before sunrise for an uphill walk to the mines, work a long shift and walk back in the dark. Ruins of Crackpot Hall [http://www.geograph.org.uk/snippet/11902] and the Swinnergill Mine [http://www.geograph.org.uk/photo/4666080] are in the hill above Keld. https://en.wikipedia.org/wiki/Keld,_North_Yorkshire

Reeth is a nice village and a pleasant overnight spot. Buck Hotel is a 18th Century inn [http://www.buckhotel.co.uk/]. Reeth is a good center for wondering around the Dales NP. http://www.reeth.org/Reeth-village-information.htm

North York Moors National Park:

http://www.northyorkmoors.org.uk/

Carlton Bank (408 masl), Crigley Moor (435 masl) and Cold Moor are a scenic traverse of the Cleveland Hills [http://www.walkingenglishman.com/northyorkmoorstopten.htm]. Stop in Carlton Bank at the Lord Stones Cafe which seems to be built into the hillside. http://lordstones.com/cafe/

Glaisdale is a nice village base for exploring the North York Moors NP. We didn't see much of scenery between Cold Moor and Glaisdale because of the driving rain, but the Lion Inn at Blakey claims to be the best pub in the North York Moors. Best or not a hot bowl of soup at the Lion Inn was a life saving interlude on the way to Glaisdale. http://www.lionblakey.co.uk/

Grosmont. Good place for a pork pie and a view of the steam train [http://www.geograph.org.uk/snippet/11079]. North York Moors Steam Railway: http://www.nymr.co.uk/

Robin Hood Bay. End of a journey. Walk down to the bay and dip your toes into the North Sea, then have a pint at the Bay Hotel. http://www.bayhotel.info/

"There's a train through the moors between Whitby and a town north of York that I can't remember now -- not a regular "transit" train but a scenic ride. "

That sounds like the North York Moors Steam Railway that runs from Pickering to Whitby. It is indeed scenic and it's nice to ride in those vintage carriages. The previously mentioned Goathland was notable for something else before Harry Potter - it's the location where the TV show "Heartbeat" was filmed (I still watch reruns every Saturday night on the local PBS station). Grosmont is another station along that rail line.

York is most definitely worth at least two (and preferably three days) as there's lots to see there. If you visit York, be sure to see the excellent Rail Museum (free, as I recall) which has an incredible collection of trains including the Mallard which set a world speed record for a steam train (I believe it's still in running condition). If you're interested in WW-II history, you could also visit Eden Camp which is short distance north of York. It's been voted "Best Tourist Attraction" a couple of times.

North Yorkshire is one of my favourite parts of England.

Thank you for all the great comments and suggestions. We will certainly take them into consideration as we begin the planning of this trip. I think we will take the train from London to York and then rent a car in the area to see the area beyond the city. I appreciate all of your wisdom.