Get a subscription to the trade newspaper Backstage, or the website version at the Backstage website. Here, you will find lots of resources, including a large section for Equity and Non-Equity auditions in New York.
Know the difference between Equity and Non-Equity auditions. If you don't have an agent, you are probably Non-Equity. However, you do not need to have an agent to get your Equity card, and you do not need to have your Equity card to get an agent. Even at an Equity audition, Non-Equity actors will be seen at the very end, once no more Equity actors are available. With a Non-Equity audition, everyone will usually be able to make an appointment, no matter what your union affiliation may be.
Send headshots everywhere! You never know who might be interested in bringing you in for an audition. Send a headshot with your resume stapled on the back to anyone who is holding auditions for anything listed in the print or online versions of Backstage. Even if you do not feel you are exactly right for the role, the only thing you have to lose is postage.
Check the Playbill website for open calls. On the "Jobs" section, directors and casting directors will sometimes list a role they have not been able to fill, or information about an open call. An open call is when anyone is invited to audition, in order to fill the talent pool for that particular company or production. Open calls are a great way to get to meet casting directors and are either by appointment or first-come, first-serve.
Be persistent! Even if no one calls at first, keep auditioning and sending out headshots. Sooner or later, you will be asked to attend a callback, or even better, be cast in a great role! Keep on putting yourself out there. The more aware of you directors and casting directors are, the more likely they will be to cast you!