FindACase is a simple, clear case-law search engine that goes back to 1955 for New York cases and 1886 for the US Supreme Court.
Google Scholar has an effective case search feature that lets you limit by jurisdiction and by time range.
Here is a useful blog entry that goes into more detail on free sources of case law.
There are three levels in the federal court system. Trials are held in the federal district courts. If the losing party at trial brings an appeal, the appeal is heard in a federal circuit court of appeals (often referred to as the circuit court). Here is a nice map of the district and circuit courts. Manhattan is in the Southern District of New York, abbreviated as S.D.N.Y. Brooklyn, Queens and Long Island are in the Eastern District, or E.D.N.Y.
All of New York State, along with Connecticut and Vermont, is in the Second Circuit Court of Appeals, abbreviated as 2d Cir. The circuit courts are numbered 1 through 11, but there are also a couple of non-numbered circuit courts. The most important of these is the District of Columbia Circuit (D.C. Cir.), which hears appeals of cases that challenge federal regulations or executive actions. The DC Circuit has been a major source of Supreme Court justices.
The U.S. Supreme Court hears appeals from decisions of any of the federal circuit courts. Uniquely in the federal system, the Supreme Court can choose which cases to hear and which to refuse to hear. One frequent reason for the Supreme Court to decide to hear an appeal is a conflict over a legal issue between two or more circuit courts.
Citations to federal cases usually have only three parts, the names of the plaintiff and defendant, the reporter where the case was published, and a parenthetical showing the court and year of the case.
Federal district court case
National Collegiate Athletic Association v. Christie, 926 F. Supp. 2d 551 (D.N.J. 2013).
The Federal Supplement is the official reporter for the federal district courts. This case appears in volume 926 of the second series (they don't like to go beyond 999), starting at page 551. The case was decided by the federal district court for New Jersey in 2013.
Federal circuit court case
Makaeff v. Trump University, LLC, 715 F.3d 254 (9th Cir. 2013).
The Federal Reporter is the official reporter for the circuit courts of appeal. This case appears in volume 715 of the third series, at page 254. It's from the Ninth Circuit Court of Appeals, based in San Francisco.
Supreme Court case
Tinker v. Des Moines Independent Community School District, 393 U.S. 503 (1969).
The official reporter for the US Supreme Court is called the United States Reports. Since it covers only the Supreme Court, there is no need to specify the court in the citation.