Start your research in the local airports in NY and NJ. Also, keep in mind, if you are going to be doing CFI, you're in the northeast. When the snow storms come rolling in, you're not going to be flying. If you're looking to build time or have year round flying, consider FL, AZ, or CA.
NYC/LI is expensive. If you decide on LI, you'll need a car. Public transportation will leave you with at least a mile walk from the train station. Consider your commute. During the summer weekends, most highways on LI are congested unless you leave early in the morning. During the week, you'll have commuters between 7-10am and 3-7pm slowing you down. And the NY State Police have nothing better to do than to give out traffic and speeding tickets on the highways. If I were you, I'd avoid LI. Live in NYC and hit up the NJ airports or KHPN.