For more recent cases, the Supreme Court’s website is an excellent place to start, as it provides information on cases pending on the docket, schedules of oral arguments, slip opinions from the Court, and other relevant information. You may also wish to consult US Law Week, which publishes a record of the Supreme Court proceedings and other developments, and the SCOTUS blog, which tracks the activities of the Court in both editorial and statistical reports.
LexisNexis: United States Supreme Court Transcripts (Oct. 1979 – date)
Westlaw: Classic database SCT-ORALARG (Oct. 1990 – date)
Oyez - RealAudio recordings and MP3 files for selected cases.
United States Supreme Court – oral argument transcripts for cases argued from the October Term 2000 to the present.
Landmark Briefs and Arguments of the Supreme Court of the United States. Location: (L5) KF101.8 .K87. Select transcripts or summaries of oral arguments published roughly two years following the decision. (1793 – date)