Frequently, students and faculty desire to research a case. This may mean different things. Those interested in decisions as a matter of law are best served by the relevant case reporter for appeals of a lower court's decision (see below). Those interested in the documents around a case at any level under any jurisdiction are engaged in a different research process. Much of what they do next depends on when and where the case occurred.
There is no one-stop site or resource for these sometimes voluminous -- sometimes scanty -- sometimes entirely missing -- case materials. If they survive, briefs, transcripts, and other materials likely are at a local, state, or federal archive. The librarians of Yale Law School have compiled an excellent guide for those who are intent on research involving court records, briefs, and oral arguments. Consult your liaison librarian for additional information.
HeinOnline provides a detailed guide for its Supreme Court Law Library. It also serves as an excellent introduction to the materials available for the Court's activities.