The first stop east of the Connecticut is Niantic. I have several friends (two with kids) who dock there and like it. Getting in and out under the bridge in a fairly narrow channel can take some attention.
Next east is New London. Lots of marinas on your port side as you enter the harbor. I tied up at one of them for two years and finally left because of the wakes from the ferries. A little further in there is Crocker's Marina on the left but you have to go under the railroad bridge (normally open) to get to it. Nice place with a pool but no trees or shade. They have fuel, by the way.
Next down the coast you are in Fisher's Island Sound. A few miles east and you can enter the Mystic River and more marinas than you can count. Some have nice facilities (like a pool) for kids. This is definitely one of Connecticut's most popular docking areas. Running up and down the river can take a while, since there is lots of boat traffic. Most people I know who dock on the river love it.
Next stop east is Stonington, Connecticut's perfect little waterside village. The only marina for bigger (30+) boats is Dodson's at the end of the harbor. Not sure about seasonal dockage there but there are lots of moorings and a launch service. No pool at Dodson's.
If you go further east, you'll be in Rhode Island.
If I've missed some good spots, someone here will let you know, I'm sure.
Full disclosure: I dock at Norwich, 12 miles north of New London. Beautiful marina with a pool and very kid-friendly. You do have to do that 12 miles up river and that turns some people off but you are docking in fresh water, which is an advantage.
Bill