Introspection is one of the most fundamental necessities of trying to understand who you are and what your place in the world is. It should be necessary to everyone to explain to themselves in a satisfactory manner a) why they believe in what they believe b) is there a possibility of them being completely and utterly wrong in their conclusions. In addition, being able to examine your own internal process from a non-involved vantage point while it's happening is extremely helpful in creating a complete idea of your self-identity.
2. A sense of internal pluralism
The mental landscape of the human mind is not a singular thing, it can be best described as a debate by an inconsistent committee of contradictory opinions. I dare say that most people don't realize that they have more than one internal voice, especially since it's considerably easier to go along with the conclusion of the most vocal one at any given time. Just recognizing the fact that you do indeed have, as it were, an angel on one shoulder and a devil on the other, helps to give you a sense of who you really are.
Just for clarification, I'm not talking about hearing voices. I'm talking about the fact that there are different parts in a person's mental make-up, otherwise there wouldn't be much sense in the idiom "to argue with oneself" or in the concept of self-doubt. The non-involved vantage point that I mentioned earlier basically means that a part of you notices when you're arguing with yourself and can observe the process.
For any of this to make sense, every adult person should have a satisfactory rational explanation as to why they can say that an external world beyond their own internal world exists in the first place. Without having done so, one's opinions on the external world seem rather pointless to begin with, so it is an essential foundation to build everything else on.
4. Introspective illusion
We humans tend to trust our own introspection to a greater extent than that of anyone else, because we have no direct means of observing the latter. What this really means is that we tend to evaluate our own actions based on our underlying internal motives, and everybody else's based on the consequences of their actions. The net effect of this can be devastating, as thinking along these lines makes it impossible for us to appreciate the internal motives of anyone else. To give a prosaic example: if you slip on, say, a wet surface, you'd think to yourself "it wasn't my fault that I slipped, the conditions were surprising and unfavorable" but someone observing the incident might simply think "whoa, that dude is really clumsy." When you extend this to a confrontational situation, you end up with some of the bloodiest conflicts possible; people only realize their own justifiable motivations for aggression, misunderstanding, or simply not caring about those of of the opposing party. This results in rhetoric like "we are simply trying to defend ourselves from an external threat (own internal introspection), which was originally instigated by Those Evil People because they're a bunch of Really Nasty Bastards (simplification of introspection of other party)." The results of this kind of thinking can be seen in every genocide that has ever befallen our species, because just observing the consequences makes it a lot easier to label someone evil. This mode of thought is endemic to being human, but at least you should have a mental warning signal going off in your head when you notice yourself doing something like that.
One more example to illustrate the point; you lend your car to a friend, who ends up accidentally crashing it. He/she'll be saying how sorry they are and how they didn't mean it, but you'll still be angry, because f**k, you crashed my car. If the situation were reversed, you'd be exhorting how you didn't mean to do it, but the person from whom you lent the car would be pissed off, maybe even more so because of your 'excuses,' and you'd wonder why he couldn't relent even a bit, because you really didn't mean to do it. The reason is quite simply because your explanation has to do with your motives, but his/her perspective is based solely on the consequences. You have fundamentally two completely different perspectives in that given exchange. Sounds quite familiar when put like that, doesn't it?
Once you start with introspection, and realize the possible fallacy that you're unwittingly committing by downplaying those of other people, you quickly run into the possibility that everything you think and believe might be utterly and completely wrong, or at least not as absolute as you previously thought. This usually results in either taking a healthier perspective regarding your own opinions or a full-blown existential crisis.
A friend of mine once said that you can't really call yourself an adult before having dealt with the idea of existentialism, and I agree completely. We humans have an unbelievable knack to ignore the abyss beneath the thin shell of our own psyche, and that can lead to acting out of sheer ignorance. What I mean by this is that if you never even give a moment's thought to the possibility that there are no absolute truths in anything, or worse yet, you get scared by the very notion of it and avoid the issue, you tend to cling to things which proclaim to be absolute truths. This is nothing short of sticking your head to the sand until the nasty thing goes away, and you're just as likely to get bitten in the ass if you do so. The Nietzschean version has been over-publicized to some extent, not giving enough room for various other thinkers. My personal favorite is a Norwegian philosopher by the name of Peter Wessel Zapffe, whose essay "The Last Messiah" encapsulates the concept of existentialism in a slightly "healthier" way than Nietzsche's body of work.
Epicurus was a rather amazing Greek philosopher, who, through sheer logic, could come to such fundamental conclusions that it took almost 2200 years for science to develop to prove them correct. The basic idea of Epicureanism can be encapsulated by the Tetrapharmakon: Don't fear god, don't fear death, what is good is easy to get and pain is easy to endure. Especially in light of the existential fear of death that we as human being tend to generally share, the idea is very simple: when you're alive, you're alive, so thinking about your own death is premature and pointless, and when you're dead, you tend to be too busy being dead to notice it at all. A lot of Epicureanism can be seen as a precursor of the aforementioned existentialism, which he, again, predated by more than 2000 years. Epicurus was highly influential in the imperial Roman period, but the Catholic church did an excellent job in getting pretty much everybody to either forget or completely misunderstand the school of thought, as it was in some respects diametrically opposed to certain tenets of the Catholic faith (especially the relativism inherent in Epicureanism doesn't really work with any strictly dogmatical system of belief).
8. Logical fallacies
Especially in a democratic society, one should always have a critical mind regarding statements given by someone promoting any given solution or opinion. Logical fallacies, both formal and informal, are used as much today as they were in ancient Greece or Rome, where they were first codified. Just being able to spot a post hoc ergo propter hoc-argument or understanding the concept of onus probandi will give you a better view into the rationality of the opinions and arguments of others. This is simple enough with external views and opinions, but the really hard part is to apply the same rigorous and stringent standards to your own thinking. This is a part of the introspection mentioned in the beginning; to be able to do that properly means that you notice when you're taking the easy and intellectually dishonest way in an argument, and to preferably decide not to do so.
9. Hume's guillotine
Be it due to lack of wont or that of capability, distinguishing between normative and descriptive statements is something people normally don't really do properly. This is the origin of a lot of rather awful argumentation, because people mix what ought to be and what actually is. This is important, because to be able to say that "this is the situation, it would be better if it were so" implies that you have an underlying rationale to evaluate the current situation and how the suggestion you're making would make it better. This further implies being able to understand the rationale behind how you formulate opinions in the first place, which requires introspective understanding of oneself on a level that a lot of people really can't be bothered with.
Jeremy Bentham, at least to me, was one of the greatest thinkers ever to have lived. His concept of utilitarianism gives a strong teleological argument how to evaluate and formulate normative statements, which is simply: maximize utility (i.e. as much good as possible for as many people as possible). This idea underlies all modern economics, and it is one way to answer the dilemma posed by Hume's guillotine. One should however understand that increased utility doesn't necessarily mean "more money," otherwise it'll be quite difficult to understand a lot of how the world actually works. In some ways, the forerunner of utilitarianism was, yet again, Epicurus.
FP EDIT: I posted this late last night and discovered I completely forgot to add the source. The original writer @Alexagon contacted me and asked to add username only and no link the original article. If you're wanting the article source please contact @Alexagon.
I'll now perform rituals/sacrifices to the Internet Gods to ask forgiveness for forgetting the source!