I grew up in Montville, just north of New London, and have lived in Norwich, Waterford and New London proper.

If you're used to urban living (it sounds like you lived in some decently dense cities in the UK) and want to be able to walk to literally anything, and enjoy things like coffee shops and live music, just move to New London. It's a cool place and the only thing resembling an actual city in this part of the state, considering how hollowed-out and economically depressed Norwich is.

If you're moving to near downtown, try to stay east of Jay/Huntington Streets. If you're looking for a more suburban residential area things are pretty nice south of the hospital. Those aren't ironclad, of course, but it's probably a good place to get started with your search. Depending on your tolerance of crime/vagrancy/squalor you can expand your search from there.

Montville, Waterford, Groton, Ledyard, East Lyme... meh. They're all pretty much the same from where I'm standing. All perfectly decent car-reliant suburbs.