Thanks to all. To channel Rumsfeld, it's interesting to move items from the "unknown unknowns" to the "known unknowns" category. For example, I didn't know that the Oyster card came in visitor and non-visitor editions.
In my experience, the main issue is finding a place to put your teeny-tiny U.S. SIM card where it won't get lost while you're on your trip.
I have run into the issue of storing a SIM before, so I believe I am prepared to deal with it. On the other hand, sometimes issues find ways around my preparations. If ever there will be a time for a SIM to wander off, it will be in another country.
The [Piccadilly] line is all stops. If you are likely to be tired on arrival, you may want to consider the Express from Heathrow. It is expensive, but I believe you can buy ahead on the internet for a better price. But if you are arriving in the morning and will have to wait until your hotel room is ready you may as well take the Picadelly line for a fraction of the price.
I have visited London four times and love it more each time. A few tips: travel light, most stations have lifts, but some do not. We found this when we needed to change to the Circle line at Paddington. That small section did not have a lift. The first time we lugged heavy bags up and down flights of stairs, the second time a helpful staff member pointed out that we could take the next train one stop in the opposite direction, just walk across the platform there and catch the next train back. It worked!
Be prepared to walk, if possible. Both sides of the Thames River have excellent walking paths along the banks. and you see so much more. Be aware tube stations may have more than just the London Underground lines and other lines may get you somewhere faster.
Re Bath: we have driven there and parked in the visitor parking around the perimeter, at a cost! It is a compact place and from memory the train station is close to everything. I recommend the hop-on, hop-off bus to get around. You will see lots and get off at the places you want to visit. There was a long queue for the Baths, maybe plan to get there early. Well worth seeing.
My only concern about the Piccadilly Line is having the bags take up space on the car. Carrying the bags should not be a problem, and I'm hoping that by time we clear customs and orient ourselves after our 0800 arrival, rush hour will be over.
I didn't follow the description of taking a train one stop and then catching the next train back. Did the next train back take you to a different station? Is there a lift on one side and not the other?
We used to be walkers, and I'd like to think we still are. We do expect to walk unless it would take too much time.
In the evolution of the plans, the current thought is to take the train to Bath.
I forgot to mention the very useful www.seat61.com. Good for down to earth info and hints, also good prices for travel in Britain and Europe. I have asked questions there and received prompt answers.
I am pretty sure Vauxhall is in Zone 1, if this has not yet been covered.
Another thing about travel in Britain, France too - public toilets are pretty scarce. They are commonly to be found in cafes and restaurants, so take advantage of any you find. Sorry to be so basic, but I find it a really irritating feature.
I have read pages at Seat61, but never interacted further. Thanks for the info.
There is no need to apologize for providing basic information—especially about toilets. In foreign countries, I carry my own toilet paper and suggest others do the same. Happily for me, I had already begun this policy before I entered the public restroom in Rome where one is expected to tip the attendant in advance. I don't know if I didn't tip enough or if the locals had different digestive tracts, but the three squares of toilet paper that the attendant gave me would not have been enough.
Thanks again for all the guidance.