Once you know your precise question, it's time to hit the law library. The first thing you're going to look up is statutory laws that pertain to your circumstances. Statutory laws are those laws written and passed by government legislatures. In cases where there might be conflicts between them, federal statutory law overrides state statutory law, and state statutory law overrides common law. (Tort laws might be what pertain to your case. These and contract laws are what define civil liability exposures. Tort law is subdivided into areas of negligence, intentional interference, strict liability, and absolute liability. In our neighbor's goat instance, this might turn out to be a tort law case of negligence on your neighbor's part. But do your research first.) You'll want to look up Case Laws. Within these you need to find a statute or a few different statutes that apply to your particular circumstances.
Once you have these down, the next part of the law library you will turn to is the West's Digest System. In this system, cases are summarized by legal topic. So you can now use the material from Shepard's to find even more relevant case law material through cross-referencing.
Now, once you have all of your cross-referenced cases in tow, you need to make sure that your citations are not out of date and would still apply in a court of law. Your state's congress might have revised or reversed the statutory law in question, and if this is the case then you might not have any case at all. So, look up your cases in Shepard's Citations for Statutes again and see if there has been anything to overturn the laws that you've been researching and planning to use to make your case on.
If you take this approach, you'll be able to move as quickly as possible with the highest accuracy possible in any law library. Once again, however, you have to first have a well-defined, focused legal question. You should not be moved by too much emotion, even if your case is definitely an emotionally charged one. You must start off with a rational approach, and your first purpose in going to the law library is to see if you actually have a case to begin with.