The United States has a hierarchical court system. Cases are heard in a trial court and then may move up to an intermediate appellate level, and then a final appellate court.
Trial court appeals are heard by appellate courts, such as the U.S. Court of Appeals, or a state Court of Appeals.
The final appellate court is the U.S. Supreme Court or a state Supreme Court.
The Chicago-Kent College of Law provides a multimedia archive, including audio records, of the United States Supreme Court, called the Oyez Project. The website allows you to search by date, justice, and topic.
Please remember that case availability is limited, particularly in the lower trial courts. Not all cases are published in reporters and while many courts do have websites, they often only provide access to the most recent cases.