This collection of lecture notes used for an intro pure math course also covers communications in math like how to write and structure a proof https://infinitedescent.xyz/
There is also classic material, like Descartes 'Discourse on the Method'. As for being able to argue well, any books on thesis defense will help like what a committee looks for when tearing apart your hypothesis and results/logic.
Coursera's course "Learning How to Learn"
Thinking Fast and Slow, Kahneman
How to Read a Book, Adler[0]
The Trivium, Sister Miriam Joseph
How to Solve It, Polya
I'd also recommend others that have already been recommended, such as the Discourse On the Method, the logic course (which I haven't personally taken, but a logic course certainly helps enter the right frame of mind for evaluating arguements with others or yourself, ime) and of course, learning more mathematics.
[0]: There's also an excellent summary here: https://pastebin.com/wGFMM1pZ
Unfortunately I don't have any clear cut answers, and sadly I don't think reality does either - by nature, it's a question that is subject to personal opinion and biased answers. But if you find a satisfactory answer, please let me know. And as a final note, I think I've learned the most from debate - be it online or in person. Engaging in challenging discussion whenever possible (figuratively). It may sound lame, but forcing myself into such discussions, or at least reading them carefully and analytically, has helped me tremendously.
https://en.wikipedia.org/wiki/Trivium
These are good areas to pursue IMO.
I've always enjoyed conversation with people that look at the world from a different perspective. Its interesting to gain some of their perspective and it also helps me look at the world from different angles.
I usually I commute 1h10m a day, by bus, and many times I just think about everything. Sometimes I have to shutdown my phone, or make sure I have no music, 4G, videos, ebooks, so I don't get distracted.
You can think about your life, future, past, work "puzzles", economics, politics, friends, family, something you forgot, make comparison between them, etc...
Also check out LessWrong and "Rationality: From AI to Zombies" - the best book on "thinking" I have ever encountered.
So my question to you is, why do you want to think? To have better relationships? To be better in your career? To be a better musician? To some extent you’ll need logic, empathy, and intuition to succeed in life but you’ll need a lot more logic as a software engineer, and a lot more empathy to be a social worker.
Learn to identify the assumptions of a position. I have a colleague who is a diehard Young Earth Creationist. He pointed me to some videos by Ken Ham (I believe). I watched them, because I wanted to understand the position. It's been a few years so I can't recall the precise flaws in the arguments, but here was the structural problem: Everything he said, ignoring how he opened an argument, was entirely logical given the initial assumptions. The entire problem was the initial assumptions (a flawed understanding of genetics, physics, and chemistry). If you took his assumptions as givens, then everything he said was quite persuasive. If you, however, actually have studied those fields or have the ability to study them, you can quickly identify why his conclusions are false.
Nearly every debate you have with someone, you will find that the disagreements start with the premises and you have to identify these. From there it's an application of logic to see how they get to their actual stated views. With this understanding, you can determine whether you can even change their view. Their assumptions may be so ingrained they cannot change (like my colleague), or you might find your own conclusions changed (develop a better understanding of the history of some country and why that places them in some position today, for instance). But you'll achieve change via logic and not emotion.
Sometimes, though, it does come down to emotion. I had some moderate success in getting a colleague to care about prison conditions by appealing to his religious views and pointing out the conflicts between claiming Christianity and also thinking it's ok for men to get raped by other men in prison ("ok", he felt no sympathy for them and thought they were getting what they deserved). I still used logic in that discussion, but one of the premises was his faith and the value he placed in it.
Other math (algebra, calculus, stats) can be proof heavy, but may require a higher level of math/logic sophistication to get into (you'd be looking at textbooks targeting university juniors and seniors, maybe even grad students). Whereas introductory geometry often includes proofs immediately, as do set theory and discrete math.
Good luck in your efforts.
1. Learn to remain silent.
2. Try to Observe thing.
3. Go to Bathroom, Sit and talk to your self [Improve talking by stopping to speak]
4. Repeat and Rise.