Historically, the term "reporter" described the individual person who compiled, edited, and published volumes of case law. Now "reporter" refers to the published books containing judicial opinions. Below are three sources where you may access reporters of United States Supreme Court opinions.
How do you find the docket number for a case? You can search https://www.oyez.org/ using the party names (ex. Roe v. Wade).
Why search by docket number? You can search by party names, but sometimes this will lead to confusion. Docket numbers are unique to specific cases.
Note that there is an approximately 5-year delay between the date of a Supreme Court decision and its publication in the official U.S. Reports. The Supreme Court publishes Slip Opinions and Preliminary Prints long before bound volumes of U.S. Reports. During this window of time, you may cite to the unofficial reporters listed below, which publish much more quickly.
Records and briefs are the papers submitted to or generated by a court in a particular case, including complaints, motion papers, court orders and briefs filed by litigants and other interested parties. Availability varies for records and briefs pre-dating digital preservation, but Nexis Uni provides access to a large portion of the available Supreme Court records.