Edit: Oh, you added more. Well, Ranking of Kings is the only one I haven't watched out of these, and I've heard great things about it. I should give it a try. Thank you.
I had the same problem before watching Barakamon. Saw Handa-kun without realizing it was a prequel and didn't really like it. But I would absolutely second the recommendation to watch Barakamon!
I watched 2-3 episodes of it a while ago. It's 100% something I'm interested in continuing, I just wasn't in the right mood back then. Thank you for the recommendation.
Yama No Susume/Encouragement of Climb is a CGDCT series, about the characters' growth as they get better at hiking and tackle more challenging mountains. Also great because of portraying real-life Japanese hiking locations. Start with the 2nd season because it's in a more traditional 30 minute episode format.
I've added Dumbbell Nan Kilo Moteru? to my PTW. It was actually Masamune-kun's workout-addict personality (Masamune-kun no revenge) that made me start working out myself, so I'm somewhat interested in watching another anime with characters who work out.
I hadn't heard of Yama No Susume, I'll go read about it.
Sorry if you've seen these already, went down my list from the highest scores so most of these are fairly popular.
Dororo... in a way? It's a story of one of the main characters trying to get his body parts back to become human.
Mob Psycho 100 kind of fits the bill, while self-improvement is not the focus, it's the driver of the story and the motivation of the protagonist.
Inu-Oh - kind of a similar premise as Dororo, but in a movie form, and instead of slaying demons it revolves around becoming a human via music performance.
A recent one: Onii-chan wa Oshimai (OniMai) - the main character, a NEET, is turned into a younger aged girl and begins their journey to become a functioning member of society.
And last, Serial Experiments Lain. This doesn't directly fit, but it's very philosophical and tackles themes as identy and more. Mindfuck incoming thouhg
Vagabond is about a ruthless samurai warrior going on a journey to become "Invincible Under the Sun", eventually coming to learn about the value of life and what he truly wants out of his life. It's a masterpiece of a story that I think everyone should read.
You may have already seen it based on your other responses, but Hibike! Euphonium is a perfect fit.
I'll add something you probably haven't seen: Go! Princess Precure. It looks at the struggle of achieving your dreams in a nuanced way considering its target audience.
Plus, both of these look incredible. Hibike is KyoAni's signature character animation, and GoPri is an action-sakugafest.