This 2363-acre park, at the southern tip of the eponymous island, is full of wildlife and has a nature center and a beautiful swimming beach. To reach it you'll need to cross the Thousand Islands International Bridge (toll $3).
Camping options here are excellent, with riverfront tent sites, cabins and family cottages. Be mindful of phone usage on the island: its proximity to the border means you might accidentally pick up Canadian cell service.