There are three airports serving New York City area – JFK, EWR, and LGA. Most major airlines provide flights to these airports, including, but not limited to, Cathay Pacific, Emirates, Singapore Airlines, EVA Airlines, and Etihad Airways. The cost of getting to these airports is usually the cheapest as compared to the cost of getting to Ithaca-Tompkins Airport or Syracuse-Hancock Airport directly. To get to Cornell from these airports, there is one recommended option:
Take the shuttle bus from your airport to Port Authority Bus Terminal in Manhattan (around 15-20 dollars one-way). The layout of the bus terminal might be confusing, so please go to their website (http://www.panynj.gov/bus-terminals/pabt-terminal-map.html) before flying to NYC to save time at the terminal. Buy an Ithaca-bound bus ticket at the Shortline Bus counter on the second floor, and board the bus (ask the counter agent about which gate the bus will depart from). There is usually one bus departing every two hours, and the fare would be around $40-45 one way. The ride will take approximately five hours – the bus will transit in Binghamton first – and get off at either Collegetown, West Campus, or North Campus (do not get off at the Ithaca bus terminal!) Congratulations, you have arrived at Cornell University!
If you have any relatives staying at New York City, it is absolutely preferable to ask them to help you navigate the city and the bus terminal. If you have no previous experience with the subway system in NYC, please do not take the subway, especially since you will be carrying so many things with you.