You already know many reasons to care about improving your emotions or being more aware of them. A reason I’d highlight is that emotions commonly lead to people being defensive and biased when trying to have rational discussions.
Like most of your more intuitive ideas, emotions are generally pretty in line with traditional and conventional knowledge, and shouldn’t be ignored. If you’re going to make changes, you should first seek to understand your emotions in a non-judgmental way. Then, as with any tradition, try to understand what the point of it is. What benefits does it provide? What problems does it solve? If you don’t know what the upsides are, you aren’t yet in a position to replace it with a superior alternative.
Never assume your emotions are fully wrong or irrational. Instead, they should be treated like an idea that has a point, reasoning and arguments. You can have an internal dialog using rational arguments, just like you would when two of your explicit ideas conflict. You should look for win/win solutions to resolve a conflict between an emotion and a different idea you have; don’t suppress your emotions or assume their information is false. Your goal should be to figure out better ideas, with no important downsides, that satisfy all parts of your personality; don’t assume at the start what the conclusion should be.
You can try to change actions you think are bad, like yelling at your spouse. And you can try to weaken emotional reactions you think are bad. But you should also try to understand yourself and why you’re feeling the emotions that you do. You shouldn’t try to get rid an emotion entirely until you understand why you’re feeling it, and understand what that part of you wants or values, and then figure out a solution that you can be fully satisfied with. Don’t just try to win a war to destroy part of yourself or suppress part of yourself with willpower.
Often, when you have negative emotional reactions, even if you’re wrong, you’re partly right. E.g. the thing you’re mad at your spouse about is actually partially bad, so you shouldn’t yell or get angry, but they should change something too. Or you have a problem that should be solved instead of dismissed, and your emotions are alerting you to the problem. E.g. you’re looking at something good in the wrong way, so instead of just suppressing your negativity, you should try to understand it better and change your thinking so that you see why the thing is good.
In Emotions, I talk about learning to be calmer and to take your time reacting to things. If you can slow down enough to think about what you’re doing, then your immediate emotional reactions won’t control you.
In my Bounded and Unbounded Emotions, I talk about putting boundaries on negative emotions so they aren’t fully out of control.
The first step towards self-improvement is being able to non-judgmentally introspect. You need to pay attention to what you feel, when, and why, and focus on understanding yourself without attacking yourself or assuming your existing emotional ideas are bad. If you’ve recognized a problem you’re having, you should be optimistic that it’s solvable somehow, but don’t assume that your emotions should bear most of the blame.
There’s potentially helpful advice about non-judgmental introspection in the self-help book Allen Carr's Easy Way To Stop Smoking, which talks about changing how you feel about smoking by changing your ideas. I liked the book even though I never smoked, and it’s possible to apply the ideas to other topics besides smoking, but it requires skill and effort to do that.
Some people find mindfulness meditation helpful for learning to introspect better. It can be treated as a learnable skill rather than as (Eastern) mysticism. It involves learning to mentally step back and examine your ideas as objects that you can think about, instead of just experiencing them directly or immediately. All information about meditation has flaws, but some has useful parts. A good place to start is Sam Harris’ Waking Up app with guided meditations, or the book Meditation For Fidgety Skeptics. There’s also Medito, Headspace and Balance. These meditation recommendations come from a friend of mine.