For most federal court filings, particularly anything at the appellate level, you may be able to find pleadings on Westlaw or Lexis. You can search briefs directly here on Westlaw and here on Lexis. Or, if you are looking for filings associated with a particular decision/case, find the opinion on Westlaw and click the Filings tab at the top of the case.
Similarly, on Lexis, once you have found your case, look on the right under Related Court Materials.
If you don't find what you need on Lexis or Westlaw, head over to Bloomberg. Once you have signed on, look under Popular Links --> Searches --> Dockets. This will allow you to access almost anything filed in federal courts. Keep in mind that while you will not personally incur financial charges, the law school does have a limit on how many documents we as a group can access and download. So, please use some caution in getting documents from Bloomberg Law.
Use the Lexis and Westlaw methods above to find state court materials -- but keep in mind that very few will be available. Bloomberg has virtually no state court materials available under our subscription. You may be able to find briefs and other pleadings filed by advocacy organizations in state court, if the lawsuit is of current public/political interest, just by doing some basic Google searching. But often, state court filings are not available unless you are willing to travel to the courthouse and/or pay hefty fees.