Hi - what an exciting adventure.
I know sutton fairly well so can pinpoint the good and bad areas to live. There is an adequate shopping centre there although you may want to venture to Kingston or Blue Water for the better High end shops.
Kingston isn't too far from Sutton and has lovely riverside walks with Hampton Court nearby for English historic culture and lovely gardens.
Cheam is right next to Sutton and has a lovely park called Nonsuch Park which is great for dog walking. Also has a public leisure centre with pool. The local football team is Sutton United - not sure if they have a woman's league but worth checking out.
There's also a public pool and leisure centre in Carshalton which is next to Sutton also.
There are loads of lovely dog friendly country pubs which are good for dog walking nearby - I can pm you details if you'd like?
Schools seem plentiful there - it's some 30 odd years since I went to Cheam High so not sure what it's like these days. You also have Nonsuch (girls) and a couple of grammar schools nearby too.
There is a magazine called Family Grapevine which is very informative - I think you can google them online.
Sutton College (adult education) puts on loads of courses in various topics which may be a good way of making friends.
Don't forget you will still have an IRS filing requirement whilst you are in the UK - if you need a tax preparer I could steer you towards a local one who understands the requirements of US citizens in the UK and being outside of London her fees are very competitive (unashamed plug 😉).