In deciding what schools to apply to, read the material the school provides and, if possible, to talk with people who know something about the different schools. The majority of school in the New York area are psychodynamically-oriented. However, they do provide training in other clinical-treatment modalities. Because each program gets a huge number of applicants for a few places, they are extremely selective. Each year, people with good GREs and GPAs, strong recommendations and great research experience end up being disappointed.
Programs vary in the deadline for applications.
Some are due in early December. Others are due in January. You get applications directly from the programs. The APA Guide gives you addresses. Completing applications is time consuming, so start way ahead of deadlines.