I am looking for suggestions of a quintessential town to stay in with easy access to transportation to London, not too remote, something like Henley on Thames with a decent amount of stores. A place in good range for sightseeing. Does any place stand out in someone's mind?
This nearby thread discusses what I think is your question. My own take would be to stay in London for London sightseeing, so you don't spend time and money commuting. London has lots of relatively quiet neighborhoods without a much tourism but close to sights you may want to see.
But Hampstead seems to be a popular choice on the other thread, for reasons I can understand -- relatively quiet, with noteworthy sights of its own, and not far from central London.
Rome2rio shows transportation choices between Henley and London. It's not necessarily 100% correct, but note that the choices take at least an hour and cost at least $18 each way.
Maybe our UK friends will have better ideas when they wake up shortly.
You might look at Kingston-upon-Thames.
It has easy access by train into the centre but you are also close to Hampton Court and to Richmond ( which is also a nice little town.) Both are accessible by river trips.
Kingston itself has a nice market square and old church and there are good walks along the river. It also has some good pubs and restaurants.
https://en.m.wikipedia.org/wiki/Kingston_upon_Thames
Alan
Other London suburbs that you could consider are Chiswick and Highgate, both of which have their own identity.
A former work colleague of mine used to commute in from Kingston and said the service was dreadful and he rarely got a seat. It is a nice town. Other towns you could consider are St Albans, where the Cathedral is worth visiting and takes less than 20 minutes to get to St Pancras station or Sevenoaks, which is a good location for Kent’s castles, gardens and Chartwell, the former home of Winston Churchill.
If you stay in a town on the outer fringes of London, you will be going into and out of the city during busy times (Mon>Fri) with all the commuters. Henley on Thames is indeed a pleasant little town on a branch line to Twyford - where you change trains. (You will see the fast expresses shooting through at 125mph as they head for south Wales/ Bristol/Bath and the south-west peninsula. To head west on one of these, you must go from Henley to Twyford (change) and then to Reading - first stop west of London for most of these expresses.
I don't think it is a good idea to stay in one town for 3 weeks. I would consider also going elsewhere to see more of the country - but - be sure to be back in the London area for your flight home.