Hi, first of all, sorry if my English has flaws, it's not my first language.

I'm looking for a story with characters that grow psychologically, try to communicate properly, maintain good relationships and be responsible.

I read a lot of shoujo, and now I'm pretty tired of the usual "everyone imagines the reality differently and doesn't communicate properly about it, so every detail spirals into neverending conflict/drama". 🙄

I don't care what the story is about, or what type of manga it is, the only thing I'd like is that the characters are a bit grounded and realistic about human relationships.

For example, I liked Usagi Drop, because every day challenges are usually resolved at the best of the characters' capacities.

I also liked Kimi ni todoke and Strobe Edge because of the heroine's efforts and good will. Especially Kimi ni Todoke, since Sawako is always honest and hardworking.

But I'd like something more mature, if possible. 🙂

I think the "josei" category is something you'd probably prefer, if you're looking for something more mature and realistic 🙂
Here are some series I like that, to me, have an air of "maturity" about them in one way or another:

Cousin
Works by IKETANI Rikako
Chihayafuru
Hakuji
Issho ni Neyou yo (perhaps? Maybe it's not really "mature", per se, but certainly unique and quite astray from the typical shoujo)
Kirai
Cappuccino
Shisei (YOKOBABA Ryo)
The Other Side of the Mirror

Hopefully there will be something you like in there~

Go to work, send your kids to school;
Follow fashion, act normal;
Walk on the pavement, watch T.V.;
Save for retirement, obey the law.
Repeat after me: I am free.

Hmm... let's see...

Fruits Basket was quite something. ♥
Kimi wa Pet is pretty neat because the POV switches between the female lead and male lead (depending on the chapter) -- just a bit if internal dialogue here and there.
Suki-tte Ii na yo has a very interesting tone... a little angsty at times.
Honey and Clover is absolute ♥ (though I watched the anime).
Natsume Yuujinchou! ♥
Yuru Koi is short and sweet. You may like YOSHINO Aki's works.

NAKAMURA Asumiko has really neat works. My favorites of hers are: I Am a Piano and Perfect World (NAKAMURA Asumiko).

I also like ONO Natsume's stuff too like Not Simple, Gente, Ristorante Paradiso and Saraiya Goyou -- you can watch the anime for the latter two series mentioned. 😀

OH! YUMEKA Sumomo is the queen of emotionally dramatic works! Look for stuff under her pseudonym: SAHARA Mizu... like Bus Hashiru and Nanairo Sekai (excellent short story collections).

Taiyou no Ichiwaru is good too!!
Matryoshka was neat.
Guys After Marriage provides a good laugh.

I've always adored Hatsukare for being low on drama. Much of this is because the author just doesn't throw a lot of direct competition at them, but that also means the manga is more about their relationship than the people trying to get in their way.

Lots of my fav manga seem to have this. ;D

Seconding: Kimi wa Pet & Natsume Yuujincho.

Female demographic:
Please Save My Earth
Hana no Namae
Kanata Kara
Mars
Ashita no Ousama

Male demographic:
ES
Hourou Musuko
Monster
Kiseijuu
Majin Tantei Nougami Neuro

Thanks very much for all your suggestions!

I'm going to check them out and tell you which I've enjoyed 😉

Asatte Dance
Living Game

Both spot on. A pity there ain't much manga like this...

--
"They're talking to their balls." (FMP Fumoffu)