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Justia is a free site that provides limited access to select U.S District and U.S Court of Appeals Court dockets and filings. You can also try a quick Google search using the case name to locate court documents posted online.
For most state and local courts, you should check on the court website to identify the docket number of the case and see if any court docket/filings are available online. In most cases, you will need to contact the specific court to request specific court documents for your case.
Some filings may also be available from Nexis Uni and CourtListener Recap.
The Harvard Law Library has a Westlaw terminal, which provides access to legal content including some court dockets and fillings. Their Court Documents FAQ and Records, Briefs and Court Filings Guide can also provide useful sources (note that some of these may be restricted to HLS affiliates).
PACER can be used to retrieve most federal court dockets and filings, and anyone is able to register for an account. Users who download documents from PACER pay a per-page download charge of ten cents per page.