You can register online, by mail, in person at your local Board of Elections office, or even when you go to the DMV to get your permit or license.
To be prepared to register, have ready your personal information, like your name, address, date of birth, and a form of identification. Be prepared to chose a political party. You may preregister at the ages of 16 & 17.
You can go to this website to register online: https://www.ny.gov/services/register-vote
To check if you are registered, you can go to this website: https://voterlookup.elections.ny.gov/