Most New York cases are heard in a three-level court system. The trial level is called the state supreme court. Many of its cases can be found on this site. There are also specialized trial courts, such as the Court of Claims, the Surrogates's Court, the New York City Criminal Court, et al. If a court case does not go beyond the trial level, there is no assurance that the court's decision will be officially published.
Most appeals from a trial are to the New York Supreme Court, Appellate Division. This site lets on search for Appellate Division cases from Manhattan and the Bronx. Google Scholar lets you search for Appellate Division cases statewide. New York State is divided into four appellate departments. Manhattan and the Bronx comprise the First Department. The Second Department is the rest of New York City, Long Island, Westchester, and the lower Hudson valley. The rest of the state is split between the Third and Fourth departments. Here is a map.
The highest state court is called the Court of Appeals. You can search for its decisions with Google Scholar or on this site.
Trial-level cases that are published are reported in the New York Miscellaneous Reports (N.Y. Misc.) and, unofficially, in the New York Supplement (N.Y.S.). Here is a citation to a trial-court decision:
Rister v. City University of New York, 20 Misc.3d 195, 858 N.Y.S.2d 528 (Ct. Cl. 2008).
The parallel citation to the New York Supplement is not strictly necessary, but it helps in locating the decision. Since the Miscellaneous Reports includes cases from several different courts, one has to specify which court decided the case in the parenthetical.
Here is a citation to an Appellate Division case originating in Manhattan:
Baez v. Barnard College, 71 A.D.3d 585, 898 N.Y.S.2d 29 (1st Dep't 2010).
Decisions by the New York State Court of Appeals appear in the New York Reports, which is official and covers only the Court of Appeals. They also appear in the New York Supplement (unofficial) and the Northeastern Reporter. This last series is also unofficial and covers Illinois, Indiana, Massachusetts, New York, and Ohio. Here is a citation to a New York Court of Appeals case:
Medici v. Dalton School, Inc., 34 N.Y.2d 735, 313 N.E.2d 789, 357 N.Y.S.2d 496 (1974).