There are many ways to approach any problem; this is one approach.
The example problem was not chosen to fit the methodology but is rather a random fact pattern chosen to demonstrate the thought process as this method is applied to the research problem - including altering the plan and hitting dead ends.
Often in law, you are presented with facts (e.g., my neighbor has large smelly bonfires every night) and not legal issues (my neighbor is violating nuisance law by obstructing the reasonable and comfortable use of my property). Unfortunately the law is organized according to legal issues and not facts. It is your first task in any legal research problem to convert the facts you are given into a legal issue.
How To: This part of legal research is difficult without any previous experience with the law, but luckily you will usually have some idea what you're looking for thanks to your law school classes. If not, the best resource for determining the legal issue in your fact pattern is a law librarian. If you can at least narrow down your topic enough to know which area of law covers it (e.g., that nuisance is a tort claim), that's usually enough to get you started.
Once you determine the legal issue, you need to figure out the language used to describe it, known as the terms of art. Conducting legal research without these terms can be difficult. Say you are researching whether a lower court can continue to make decisions regarding a case (e.g., by amending a judgment) while the case is pending on appeal? As you can imagine, you won't get far with such generic search terms as court, appeal, and jurisdiction, which are in nearly every opinion. But once you discover the terms of art for this particular situation are divestiture and functus officio, your search will become much easier.
How To: One of the best resources for determining the terms of art is a secondary source on the topic of your legal issue. This can be a legal encyclopedia's entry on your topic, a nutshell, a treatise, etc. Nutshells are preferable because they're quick and easy-to-read, and at this stage, you are just looking for a general overview of the law and the language used to describe it.
Use the index if you're looking for something very specific, or browse the table of contents for a relevant topic if you need something more general. If you don't know which secondary source is best for your topic, just ask a librarian.
Next you must determine the relevant jurisdiction and the source of the applicable law - i.e., is it state or federal? If state, which state? And source of law - is this issue guided by case law only? Statute? Regulation?
How To: While you still have that secondary source, take a closer look and see if it drops any clues as to the state and federal question, or the statute/regulation/case law-only question. The resource will probably not tell you the specifics of a state law (unless of course, you're in a state-specific resource, such as Strong's North Carolina Index), but it will usually tell you whether there's relevant federal law, and whether it falls under the purview of a federal agency. Whether there's a relevant state statute might not be as obvious, but don't worry too much yet. This step overlaps with the next step.
Now that you roughly understand the legal issue presented, it's time to find the applicable law in your jurisdiction.
How To: The next step is to look in the index of your state's statutes. In North Carolina, that’s the North Carolina General Statutes. You now know the legal issue and the terms of art used to describe it, so use the index at the end of the statute set and look up your topic (or search the relevant database on Westlaw or LexisNexis). It should direct you to the relevant statute.
The trick here is not to kill yourself searching - not yet, in any case - because there very well could not be a statute, and looking for something that doesn't exist can drive you crazy. Instead, do a thorough-but-not-insane search using your terms of art and with your new understanding of how your legal issue is organized (for example, if you were researching the vicarious liability of an employer for the acts of his employee, then you would know enough now from your secondary source research to look under the large heading Labor and Employment in the statutes index.)
The reason there is no need to drive yourself crazy (yet, at least) is that you still have case law to look into, and case law, statutes, and regulations all tie into each other. If there is a recent, relevant case on your legal issue then it makes sense that the court would mention the statute in its opinion. This is not a failsafe, as sometimes statutes can particularly address one fact pattern but not another, relevant one. For example, North Carolina has a statute governing some nuisances, like poisonous snakes, but not others, like enormous external speakers. Therefore if you were looking for a North Carolina statute dealing with nuisances, if the case you found dealt with large speakers, the nuisance statute might never be cited even though the case talks about nuisance law.
Even if there is a statute, and you do miss it this first time around, the hope is that in the course of your case law research you will gain an even greater understanding of your legal issue which will, in turn, help you to conduct a more effective statute search when you try again later. For example, you may learn where nuisance laws are organized in the NC General Statutes generally, so you'll know exactly where to look for your specific fact pattern.
You may do the case law research before statutory research, if you prefer. However, looking for the statute first will lead you to relevant cases through its annotations. Those cases will then give you the relevant topic and keynumber in their headnotes so you can pull all the relevant cases very quickly. You can also use the citation to the statute as a very accurate search term in a Westlaw or Lexis search when you are looking for cases.
Assuming you don't find the magic statute (or even if you do, since you will still have to do case law research - the methodology will just be different), the next step is to conduct case law research in your jurisdiction.
How To: This is best done using West's Key Number System® because the key numbers are arranged by legal issue, which is exactly how you have arranged your legal question. You use the digest’s Descriptive Word Index to look up the terms of art for your legal issue. That will provide you with a topic and key number. Then look at the outline of that topic at the beginning of that topic’s section in the digest. Viewing the outline will allow you to pick the most relevant key number (the one you were given by the index may not be the absolute best). This, in turn, will send you to relevant case law on your topic.
After reading the relevant statutes and cases, it's time to formulate your best interpretation of the law on this topic in your state, then apply those rules to your fact pattern to guess how the court would rule on your issue. You then formulate your arguments, based on the cases and statutes, for why the court should rule in your favor.
How To: This is another section where you really just need to use your own brain power and legal expertise. After three years in law school you should be able to apply a law to a set of facts and make an argument. If you can't… well, that's not really a research problem.
After finding all these statutes and cases, you want to make absolutely sure that they are still relevant. Not only is it embarrassing to use a case that's been overturned or a statute that's been superseded, but it's actually malpractice.
How To: Use Westlaw's KeyCite or LexisNexis’s Shepard’s to update your research.