We are 74 and 71, with limited mobility and are headed to England for the first time, with our 20 y/o granddaughter. We have two weeks reserved in Pine Lake in Carnforth May 16 to May 30 after arriving in London on May 14. I have London plans in my mind but I'm trying to get the rest of the itinerary and transportation figured out. I know we want to visit Hadrian's Wall and in the Lake District area: Lake Windemere, Castlerigg or Swinside Stone Circle. I thought a day trip to Yorkshire Dales. I'd love to do a day trip back south to Stratford-upon-Avon to see the gardens there. Our interests include photography, local culture and food, history, gardens, and nature and our granddaughter is studying anthropology and archeology. Do we need to rent a car? We'd love to do public transportation if it's feasible or even paid tours. Any tips or tricks anyone can offer would be wonderful.
I'm struggling to get a sense of what your itinerary is - am I right in saying you arrive May 14, spend 14th and 15th in London, then on the 16th head up to Carnforth until May 30th. Do you fly home on the 31st?
From Carnforth, public transport is very limited. There are a handful of direct trains to Leeds, but for the most part visiting where you want to go will require changing trains either once or twice. Furthermore, the only trains from Carnforth are Northern ones - Northern serve small communities, so will likely stop at every stop, meaning it will take much longer to reach your destination. Your best bet would be with a car, assuming you're willing to drive on the left.
Stratford upon Avon seems like a bit of a stretch for a day trip, I'm afraid - at the very least I'd recommend going overnight. Yorkshire Dales is theoretically doable for a day, but Carnforth is coastal Lancashire, about as far away from Yorkshire as you can get in the county. While Britain is a small country, the road network in rural areas is not often very direct, and it takes longer to drive than you think to places.
If you're in Lancashire, I'd recommend visiting the Lake District, as you've mentioned, where there's plenty of lovely towns like Kendal, Keswick, Ambleside and Bowness. Also, there's Lancaster, which is an historic county town and the next stop on the line from Carnforth. For The Wall, I only know the North East end of it, but I'm sure it's perfectly feasible to visit the western end - perhaps a train to Carlisle? There is a direct one from Carnforth, and if times don't suit you can change at Lancaster, which as I said in the next stop on the line from Carnforth.