This is an exercise for analytic synthetic intellectuals. If you're not analytic, you won't get any coherent answers. If you're not synthetic, you won't generate any answers. And if you're neither, there aren't any answers for you to get, period.
See also What Are Core Values?
Theory for Non-Intellectuals
If you're not an intellectual, you don't concern yourself with questions of self-knowledge so you simply won't care about this exercise. Despite the fact that you should care as self-knowledge has many important applications in your life.
Then again, if you're a non-intellectual (to say nothing about anti-) then your whole approach to life is empirically wrong because you're assuming you can reliably predict, using a simple heuristic no less, the practical value of theoretical understanding before having acquired said understanding. You cannot and your understanding of how the world works is warped.
If you're a non-intellectual, there's nothing you can do to become an intellectual since it's fundamental. However, it remains useful to recognize your limitations instead of denying they exist. Hanging around intellectuals and following their advice when they say a purely theoretical exercise has immense practical value is a good idea.
Orient Your Life!
So what are the practical applications of knowing your core values?
For one thing, you will know what you want to do with your life. As opposed to what you've ended up doing by chance. As opposed to what people tell you they want you to do. As opposed to what people tell you you should want to do. As opposed to what will bring you the most money, the most glory, or the most acceptance.
Assuming of course that your mind and self-identity aren't so broken that you are some kind of psychopath who cares only about power. Or a narcissist who cares only about glory. Or that you haven't had your self-esteem ground down to nothing until you are so fucked up as to have no concept of self-identity period. Yes, people that fucked up exist in modern society.
It's not a coincidence that I warn against narcissism and psychopathy. The same mechanisms that drive empathy (the formation of other-identities) also drive the formation of self-identity. See Formal Definition of Empathy and Empathy, Synthesis, Autism and Psychopathy. Whenever I get around to writing them, there's still too much uncertainty in my theories.
Learn To Communicate!
For another thing, knowledge of your own and your peers' core values will allow you to understand how to interact with them. Or why you cannot. It will allow you to see why a certain line of argument will never work with someone despite it making perfect sense to you. And vice versa. It will allow you to see what's normal to you that others consider hurtful and vice versa.
And the reason why it works is that your core values are like axioms in a mathematical system. Some things can be proven in one axiomatic system but not another. Some theorems translate to other axiomatic systems. Some need to be adapted. And some will simply never translate.
So if integrity is one of your core values then logical contradictions will be intolerable. And any logical contradiction between your core values and another person's core values will mean you can't interact with them, period. Hypocrites will also be intolerable regardless of their values.
Incidentally, this kind of meta-level behaviour, where your core values determine how your core values interact, is quite common. As it should be since there's nothing more fundamental to appeal to.
Understand Your Friends!
Finally, understanding someone's core values is the essence of understanding them. You can't say you understand them without knowing their, accurate, concept of self-identity. Knowing their food preferences or their childhood damage doesn't hold a candle to knowing their life-long enduring personality traits.
The same personality traits that will explain and often even predict over 90% of their major decisions for the rest of their lives. Meanwhile, food preferences are liable to change at any time with no warning. And childhood damage tends to be overcome by analytic-synthetics just because they can. Even seemingly enduring traits like severe depression may be solved without warning.
Core values are those things a person holds which they will never want to overcome.
Determine Your Core Values
Now here's a fun little paradox. Are the people who skipped straight to this section intellectuals for not needing the sales pitch or non-intellectuals for going straight to the practical section? This will keep me up tonight, I just know it.
Okay, determining your core values is incredibly simple if you just fucking know how. Which means if someone tells you. So you can maybe appreciate that it took me more than a decade to discover mine since I only had quacks and charlatans to "guide" me. You know, the people holding PhDs in psychology and philosophy.
The questions "who are you?" and "what do you want?" are hopelessly vague as-is. In order to fully answer those questions, you need to answer: "what is the intersection of your answer-sets to the following questions #1, #2 and #3?"
- what abstract properties would you be perfectly happy were they to become universal, fundamental and inviolable laws of the universe (because they are never evil)?
- what gives you literal joy to see more of (because they are always good)?
- what kinds of people would you cheerfully murder if you could get away with it (because they have no hope of redemption)?
In other words, what abstract qualities do you feel strongly about to the point where they bring a literal smile to your face when you see them, literal tears of pain when you see them destroyed, and one of a red haze of rage, a cold burning hatred, or a spiteful contempt when you see someone who will go on to destroy them for the rest of their life everywhere they go?
Tips & Tricks
Start off with any answer to those questions. Ask yourself whether you always want exactly this with no exceptions. If you only want it 99% of the time then there may be a closely related term which is a better fit. As a last resort, look it up in a thesaurus.
Construct counter-examples. Not just to the individual core values you've identified but to all of them. Imagine the worst possible world that fits ALL of the core values you've identified. If you think it's okay then perhaps you haven't made your imagined world a bad enough hell.
If you can imagine a scenario that is obviously bad because the people in it are obviously evil, stupid and/or just plain moronic yet it obviously satisfies ALL of the core values you've identified so far then that's because there's a core value you HAVEN'T identified which it violates. This is a clue.
Ask yourself, or better yet ask one of your friends, whether the core values you've identified are sufficient to reliably predict your major decisions, your major likes and dislikes, to greater than 90% certainty. If not then that's because your real mind is much bigger than the model-of-your-mind you're constructing. It's time to start digging into some of those dark spaces of your mind.
I have identified 9 core values in myself and I think they explain pretty much all important decisions in my life. Identifying 4 core values in someone seems like the beginning of understanding them. Knowing 6 of a person's core values seems like the beginning of a solid understanding. But yeah, EACH core value must be something you have absolute 100% confidence in before it really counts.