For a Connecticut beach week, aim the campervan at the Long Island Sound shoreline and Hammonasset Beach State Park in Madison, the largest shoreline park in the state. The waterfalls and lakes of the Litchfield Hills in the northwest are the inland alternative when you'd rather have foliage than sand. Pick one. The campsite in that region is what you book first, and the van follows.
The Long Island Sound shoreline
Hammonasset is where the shoreline trip lands. A two-mile beach and a paved boardwalk run the length of the park, and at low tide the boulders and tide pools at Meigs Point are where the morning goes. You sleep in the William F. Miller loops, 558 grassy sites reached off US-1, the Boston Post Road, once you leave I-95 in Madison. Most of those are basic sites, and only 88 have water and electric, so reserve those first if you want a hookup in July. The New Haven page goes through that campground loop by loop, so start there for the site detail.
West toward New York, Fairfield County is thinner for overnights. Sherwood Island in Westport is a mile of Sound beach, day-use only, with no camping, and the closest improved state campground from there is Black Rock up in the hills. Wilton is the base on that side of the state. Farther east, Rocky Neck in Niantic takes the overflow when Hammonasset fills, and the shoreline keeps running past towns like Mystic toward the Rhode Island line. These are May-to-October campgrounds, not a winter loop.
Between the two trips, the Connecticut River is worth a detour for Gillette Castle in East Haddam, a 14,000-square-foot fieldstone fortress standing 200 feet above the water. The river camping below it is canoe-and-kayak only, with no vehicle access, so it stays a daytime stop and you sleep at a campground you can drive to.
The Litchfield Hills
Northwest of the shore, the hills are the foliage weekend. Kent Falls is the day trip, a 250-foot cascade you climb alongside on stone steps past the covered bridge, though no swimming or wading is allowed in the water. You sleep at Lake Waramaug in New Preston, 76 sites on the water with a check-in cutoff of 8:45 p.m., or at Black Rock in Watertown if you want the water and electric Waramaug doesn't have. Come in from I-84 to US-6 and US-7. October is the reason to be up here, when the lake holds the color and the peak weekends book out.
A campervan sleeps two to four. If the group is bigger, size up to a Connecticut RV rental and you can still park these state-park loops. Keep driving and the trip leaves the state. The shoreline runs east into Rhode Island and Providence, the hills run north to Massachusetts and Cape Cod, and west toward the city you're into New York and Long Island.