So, I was looking for something that portrays the inevitable difficulties and challenges on growing up and facing adulthood. As well as becoming aware and mature of our emotions and feelings about the people that surround us and ourselves. So, not exactly looking for that cliché match of love at first sight or hate at first sight that becomes love. I want something that can show in a more realistic way that maturing of feelings that we all have along the life journey. I am new into the kdrama world and Asian drama in general so, I don't know about any example to guide you. I do know an anime " Honey and Clover" ( “Nana” is a good example too) that portrays exactly what I am looking for. The challenges of finishing college, growing up, say goodbye to friends, lovers and environments to build new ones... It may have comedy, I am fine with that, but I wouldn't like it to be full comedy at all. I would prefer that it had as much of comedy as it has of drama, just like "Honey and Clover", but full melodrama will be fine too!

The first thing that came to mind is Nobuta wo Produce. At a glance, it might not look as serious as Nana, but I think it touches upon the subject of coming of age exceedingly well.

Definitely check Age of Youth! i think it exactly fits your criteria

I think that Reply 1997 and Reply 1988 could be good as well. However, they are set in the past and main characters are high shool students so it depends on your preferences. Either way both of these dramas are great so watch them anyways. Similarly but not as great (but still good) is Sassy Go Go.

I think Cheese in the Trap might fit good but at some point it gets kinda weird.

Weightlifting Fairy Kim Bok Joo is also great (but focused on sports.

Hello! Here a few dramas that you might like:

-cheese in the trap: a romance drama but at its core totally  a slice of life/coming of age story. Not laugh out loud funny but subtle moments of comedy or joy. You'll smile a lot. 

-Go back couple: A definite "nostalgia" kind of drama where the main leads revisit their youth and see things in a different light than they used to the first time. LOL moments and a whole lots of tears too. Not a sad drama though. 

-the reply series: Reply 1988, reply 1994 and reply 1997. Exactly what you're looking for though I've not watched them yet. 

-Let's eat : though all protagonists are adults, have adult jobs and leave alone they don't really act in a mature way. We see all of them grow up and learn to face challenges in a mature and responsible way. Very little romance.

-Weightlifting faiy Kim Bok joo: has all the elements you listed. Very funny and endearing characters.

-Because this is my first life

-Fight for my way

-Incomplete life: no romance if I remember correctly.

-Twenty again: Older main leads.

I love coming of age genre. I can probably just seat my weekend away with a good coming of age drama. With that said, the more realistic and compelling coming of age dramas are:

  • Last Match - this aired in the 1990's in korea so it would be a little outdated but still the themes/issues tackled like, dreams, love, future, friendshio, family were on full display
  • Asunaro Hakusho - a 1990's jdrama that pretty much just showcases a 5 friends who meet in college in their very complicated and myriad love and issues
  • Wakamono No subete - 1990's jdrama, but with most character out of school and already walking into the path of adulthood 
  • Misaeng
  • Kingyo club
  • Girls Generation 1979
  • Because Its the First time
  • Weightlifting fairy kim bokjoo
  • Reply 1997/1994/1988
  • Age of Youth 1 & 2
  • Best Moment to quit your job
  • school 2013
  • What's Up

Some of the shows I have liked in this genre are :

1. Red Teacher : Part of the drama special series-plays more like a full length movie 

2. Puberty medley: 4 eps drama - very endearing characters 

3. Who are You (School 2015) 

4. I can hear your voice

5. Pinochhio

I second all of these recommendations:

- Age of Youth
- Because This Is My First Life
- Fight For My Way
- Misaeng

On the other hand, you could also check the following:
- Just Between Lovers
- Kahogo no Kahoko
- It's Okay, That's Love
- I Hear Your Voice
- Ho Goo's Love

"Age of Youth" definitely fits your description, it's a must watch. One of my all-time faves!