Not all U.S. Circuit Courts of Appeals (or federal district or bankruptcy courts for that matter) make full (or almost full) runs of their opinions available at their websites. See, e.g. the blog post at Gallagher Law Library (University of Washington Law Schools): Circuit Courts Archiving Cites to Online Sources.
The Administrative Office of the Courts is a one place to begin when researching federal court practice and procedure, but for cases, you need to do a little detective work if you want to use a free database.
The primary method for finding federal court case documents is PACER (registration required – some fees), but opinions from the Courts of Appeals, and other federal courts, may also appear elsewhere on the Internet. The AOC has links to federal courts.
Sometimes, though, as you may have already figured out, a simple Google search, e.g. third circuit court of appeals, is the most direct route to the court’s website. (And do notice how different each court’s website looks from the others, e.g. First Circuit and Fourth Circuit.)
You will find other sources of U.S. Circuit Court opinions (see my sidebar, Legal Research-Free), though we do like Circuit Courts’ official websites when we need the opinion immediately, in part because it is the first place the opinion often appears and also because it is as close to the “official” opinion that exists – and lawyers and law librarians know that “official” is the best.
For free and low-cost legal research databases, take a look at my sidebar list. Gallagher Law Library also has a Free list, as do most law library websites. If you’re looking for state cases, be sure to check the list posted at a law library within that state, e.g. if you are researching Ohio law, look for an Ohio Legal Research guide or consult a Montana Legal Research guide, if you need a Montana case.