Well, there are no close places in the ADK's from CT. Everything will be about a 3-4 hour drive. There is the NPT, of course. But this is a week-ten days. Several short loops you can take around Peseco, and, up to Long Lake. Good paddling and quick overnighters on Forked Lake, and Long Lake, too. There are some connecting trals for the Long Trail, probably enough for a night or two. Keen and Keen Valley are both good places for entrance into the High Peaks, area. Lots of pull-offs/parking just off the Northway.
The Catskills will be a bit closer, around 2 hours. Woodlands and the surounding area has some good hikes and camping.
I would get some good maps of the area and check out the various road crossings. Almost all will have a parking spot and trail heads leading both east and west.