Public speaking helps, as does debating in person. If you want to convince others, you have to organize your own thoughts first, and put yourself in the other persons' shoes to know what would convince them. That's good mental exercise.
I would recommend personality analysis, to understand how you think and to understand how other people think. If you are going to develop yourself, you need to understand where you are first. Myers Briggs is popular but it relies on external factors to determine your personality type (e.g. what would you in this situation?), which can change from day to day, from work to home, or even minute to minute. Still, it is well studied so you can go as deep as you want on it. I prefer the Enneagram, which is less well studied, but relies more on internal motivations to figure out how you tick. There are others, I'm sure.