You can find oral argument transcripts on the United States Supreme Court Web site. Transcripts are available starting in the year 1968.
Audio files are available online beginning in 2010.
SCOTUSblog is an amazing resource for U.S. Supreme Court materials. Not only do they provide coverage of decisions as they are handed down, but they provide dockets and filings from these cases, including briefs and petitions. They also give "plain English" explanations of cases and more advanced discussions of the cases.
At Oyez.org, you can listen to many of the oral arguments presented at the United States Supreme Court.
Select the cases icon and then use the navigation on the left to view by court term or legal issue.
To listen to the argument, select a case name. The audio link will appear on the left side of the page.
To learn more about the recordings and transcripts that are available, visit their "about" page.
Although not in appellate courts, this website provides opportunities to view motions for summary judgment, jury trials, status conferences, and much more.