I love Connecticut (I live in NYC and visit friends there often, it’s about an hour on the train to where they live), but it’s definitely not a destination for a trip like you’re planning. There’s really no nightlife to speak of. There are nice restaurants here and there but you’ll need a car, and you’ll basically be spending most of the time in the car driving from wherever your hotel is to random spots. There isn’t any sightseeing other than quiet, residential beaches.

If I were you, I’d visit Boston instead. It’s not as touristy as NYC, but there’s tons of great stuff to see and experience, and it’s definitely the “real” U.S. Its super walkable city with tons of good restaurants and bars and clubs and nightlife, as well as fun and interesting things to see and do during the day. You can also do side trips to see more of the U.S., like out to Cape Cod like others have mentioned, or Salem, or Newport Rhode Island.