The Second Circuit hears appeals from district courts located in New York. The Court web site posts contact information, rules, forms, recent decisions and a practice guide called the 2nd Circuit Handbook: Everything You Ever Wanted to Know About the Second Circuit. For more information, call the Court at 212-857-8500.
The Court's Opinions database has free opinions back to at least 2002. Opinions are also available free from Google Scholar (1920s-present) and FindLaw (1995-present). For deeper databases, use Lexis or Westlaw.
Oral Arguments: In April 2013, I contacted the Clerk's office about getting recordings or transcripts of an oral argument. I was told that transcripts are not available, but I could order an audio CD by sending a letter and check for $30 to: U.S. Court of Appeals for the Second Circuit / 40 Foley Square / New York, NY 10007 / Attn: Clerk of the Court. The letter should provide the (1) case name, (2) case number, (3) date of the argument and (4) return address. For current information and questions, call the Calendar Team at 212-857-8595.
Record on Appeal: You can get copies of the case file (i.e., the "Record on Appeal") for recent cases from PACER. For older cases, copies are available from the Association of the Bar of the City of New York Library at 212-382-6746; members can get copies cheaper from the New York Law Institute (212-732-8720). If that doesn't work, call the Court's Record Room at 212-857-8620.