When I lived up north years ago, we made several trips to Letchworth State Park, which is just south of Rochester and in the Finger Lakes region. The Genesee River runs right through the park and is an absolutely beautiful place to camp, hike and picnic. The river cut up to a 600-foot gorge that have hiking trails that encircles it. There's a Lower, Middle and Upper waterfall in the gorge and make for an incredibly picturesque hike from both sides of the gorge. The Middle falls has the historic Glenn Iris Inn right next to it where couples frequently tie the knot. It's a great destination for a week long stay, which we've done several times.
The campground there is very clean and well maintained and is located towards the North end of the park. The wooded or non-wooded sites are spacious, with full hookups, bathhouses nearby, camp store, playgrounds, plenty of fields with large open spaces, and reasonable site fees.