TLDR; Acquire a solid background in law, use high quality paid non-authoritative sources, and spend a lot of time reading about the particular issue in depth.
Acquire a suitable level of background knowledge and legal skills
Firstly, you need to accept as a basic premise that law, like most serious fields, is an incredibly large one both in terms of breadth and depth. Becoming "comprehensively aware of all applicable statutes/remedies to a given situation" isn't something you will achieve easily or in a short amount of time for most non-trivial topics.
Given that premise, the starting point as suggested by Dale M's answer is to attain a university degree in law, or self study to a roughly equivalent level. The aim here isn't to learn all the law that you'll ever need. In fact, much of it you'll never use. What it does do is give you a reasonable breadth (but not depth) of knowledge across a reasonable range of common legal topics, and a certain level of familiarity and skills with legal research in general, to know where to start when faced with a particular scenario.
Yes, you could try to just learn a particular issue without a solid legal background, and this might be fine for many situations a layperson faces, but you will likely perform quite poorly with just this isolated knowledge if you are attempting any kind of serious legal work where the outcome matters. As an example of this, consider what you would do if you come across a statutory provision or a contract clause which is ambiguous in the context of your issue. You then need to be familiar with the rules of statutory interpretation and rules of contract construction respectively, otherwise you might reach the wrong conclusion (i.e. a conclusion which seems logical to you but is not the same one a judge is likely to reach). If you had only ever read up on the law for your particular issue, you may not even be aware that such rules exist. Another example of an error a layperson might make is relying on a case which reaches a strong conclusion, but being unaware that that decision is not binding in your scenario. Studying law in general to a high level takes care of many of these issues.
Start with non-authoritative sources
When faced with a new issue, my usual starting point is a high quality (which invariably means paid) legal database. These are expensive but what you are paying for is an army of lawyers whose job it is to keep up to date with the topics they oversee, and update the database when the law changes.
I use Practical Law which provides overview articles on a very wide range of issues. Lexis is another popular option. I also consult Westlaw Books which gives access to dense practitioner texts across all main legal fields.
If I'm researching a particularly complex and high value / important issue, I'll also visit a law library, take out 5-6 books for that area of law, and read the relevant chapters/sections. This is quicker than it sounds as usually the scope of the issue is narrow enough that you only need a small fraction of each book. You will get diminishing returns due to the large amount of overlap between books, but I usually find I can discover at least one or two useful points in each one.
The purpose of reading non-authoritative sources is to get an overview of the topic and, importantly, to identify as much as possible of the relevant statutory law and case law.
Note that I haven't suggested using Google. One of the first things you are taught at university is to never rely on Google for your legal research. Most articles and blogs you will find are very poor quality, often containing incorrect or out of date information, and often failing to provide decent citations that you can read and verify for yourself. It's ok to use Google to get a very quick feel for a topic and to get some clues about where to start, but you should at a very early stage turn to higher quality sources. A lawyer who solely uses Google will end up being sued by a client for negligence sooner or later.
Move on to authoritative sources
At this point it's just a matter of working through the list of statutory and case law sources you identified during the above steps.
I usually start by reading the statutes in detail and identifying the relevant sections. Often that will mean also identifying and reading relevant secondary legislation. You will be alerted to the possibility of secondary legislation by phrases such as "the Secretary of State may by order [...]" in the primary legislation. In Westlaw, you can go to that section of the primary legislation and find cross-references to the secondary instruments which were passed under it.
Do the same with the main cases you've identified. Westlaw is again useful here as it provides highly detailed cross-referencing that allow you to e.g. click through to cases which case X cites, or which cited case X.
Often a judge will very helpfully start by setting out a summary of the legal framework which is applicable to the issue, which means you quickly start expanding your initial list of statues and cases, and identify any important ones you've missed up until this point. These new additions to the list are often more helpful than the ones you identified from the non-authoritative sources. That's because the latter tend to provide broad coverage of the area of law, while cases tend to be focused in depth on a particular issue.
Starting from a section of a statute, Westlaw will also give you a list of all cases which cited that section, including those which are considered important cases. Usually a quick skim over the summaries of these cases will help you quickly identify which ones are worth reading in more detail.
This process is circular. Statutes will lead to cases, which will lead to other statutes and cases. Eventually, after spending a decent chunk of time doing this, you'll start to become knowledgeable in the chosen topic. This could take hours, weeks, or months, depending on how complex the issue is.
No matter how much time you spend researching an issue, you'll never know everything. There is simply too much extant law for that to ever be a realistic goal.