>>560182You won't be able to hike across private land, you will need to be on state owned land such as state forests and state parks.
http://www.dec.ny.gov/outdoor/41282.htmlThat site has all the info you need for camping primative.
NY is beautiful. There are lots of state parks close to Ithaca, where Cornell is. Robert Treeman is close and very nice. Letchworth State park is a little bit more of a drive, but definitely worth it. Just 30 min away is Watkins Glen state park, gorgeous, but sometimes crowded and not very /out (it basically in town, but beautiful, pic related).
If you want to be REALLY outdoors, Adirondack park covers a huge area of NY, it's bigger than Yellowstone, Grand Canyon, Glacier, and Great Smokey Mountains Nationals parks COMBINED.
Another option is to head towards Pennsylvania. The Allegheny National forest is on the NY/pennsylvania state line and covers a large area. The Allegheny resivoir is there and is awesome fishing.
Another hidden gem is Pennsylvania is the Hammersley Wild area. The largest area in the state with no roads. Near the middle is the Hammersley Pool, a cold water swimming spot miles from civilization. Backpacker magazine said it was one of the 3 best swimming holes in the Country. It's a nice hike too (maybe 12 miles round trip).
I live on the PA/NY border not far from Ithaca (near Binghamton) spend a lot of time outdoors myself.