Lots of good possibilities!
In New York state, I like Watkin's Glen and Letchworth state parks. There's a lot of hiking and exploring opportunities in the Adirondacks as well; I'm just less well versed with specific suggestions there.
In Vermont, there are many nice hiking areas. For mountains, Mt Philo (easy), Camel's Hump (moderately strenuous, with a few different trail options), Mt Mansfield (similar to Camel's Hump), and Ascutney (a little harder hike, or take the auto road and wander around the summit) are all good and popular hikes. Ascutney has a state park campground at the base, too. (Mt Philo also has a little campground, and an auto road up and down, but access for any other than a truck camper or quite small trailer would be well nigh impossible.)
There are several state parks around the Groton state forest, interconnected with a trail system that's a little closer to level. It's a pretty area. Little River state park also has some nice hiking and multi-use trails.
All the Vermont state park campgrounds have no hookups, incidentally.
For biking, there's one nice option is the Burlington bike path that connects to the longer island line rail trail, eventually heading out to South Hero (island).
The White Mountains in New Hampshire have lots of options for hiking and exploring. The Kancamagus Highway is one scenic road there with lots of pull-offs for hiking and exploration (ranging from quick strolls to significant day hikes or longer).
One resource that I find handy for New York public campgrounds is
campadk campsite photo database (which is independent of the state).