Here's some links to some campgrounds in the Finger Lakes area of NY state. I've only camped at two: Ferenbaugh Campsites and Watkins Glen KOA. That was years ago so I can't personally give an up-to-date opinion of them, but during my stay I had no complaints. Anymore I do my camping during race weekends at the track, Watkins Glen International.
Ferenbaugh Campsites -just outside Corning, NY - about 15 minutes from Watkins Glen.NY
Watkins Glen KOA Campground - a few miles south of Watkins Glen, NY and just down the hill from the track, Watkins Glen International
Clute Park Campground - in Watkins Glen, NY
Watkins Glen State Park - just outside Watkins Glen, NY above the waterfalls and gorge.
Keuka Lake State Park - near Branchport, NY on beautiful Keuka Lake
Cayuga Lake State Park - near Seneca Falls, NY on Cayuga Lake
Sampson State Park - north of Watkins Glen, NY - south of Geneva, NY - has 2,000 acres, including over 5½ miles of Seneca Lake lakefront!
Long Point State Park - north of Ithaca, NY - south of Seneca Falls - on Cayuga Lake
Taughannock Falls State Park - near Ithaca, NY - on Cayuga Lake
Buttermilk Falls State Park - just south of Ithaca, NY
Camp Elmbois Family Camping - outside of Hammondsport, NY - small campground - never camp there, but heard good things about it - on hill above Keuka Lake.
Here's a link to info and reviews of many state and private campgrounds in the Southern Tier, Central and Finger Lakes areas of NY state:
Finger Lakes Campgrounds
I'm sure I left out some campgrounds, but you'll find plenty of choices in the area.
Enjoy the beauty of the Finger Lakes area and your time there.