Hi, up until recently I've only just been watching anime and I was lost as to what to watch next. I then decided to venture into J-drama because I happened to catch a glimpse of Date and was intrigued by it. I ended up loving it a lot and frantically searched as to what to watch next. I found Dear Sister and absolutely loved it too. I'm currently in the process of watching My Boss My Hero as well as Marumo no Okite (since it's awfully similar to Usagi Drop, my absolute favorite anime of all time).

So with that said, please recommend me a J-drama/movies similar to those dramas above. My main genre would be romance and comedy with a touch of drama. As you can see I also like family/friendship theme but I try to avoid anything tragic unless it's as good as 1 Litre of Tears or Ano Hana (and I'm planning to watch Mother next). Newer drama is preferrable, as in the ones that I can find in HD but if you know any masterpiece old drama then please recommend it too. I also try not to venture into those that seems very shoujo-ish, which is why I have so far avoided Itazura na Kiss and Hana Kimi. I really love the drama version of Nodame Cantabille but that was a josei. No to the historical genre too.
Last Cinderella was really good. So was Akumu chan.
Boku no ita jikan is pretty good and is similar to 1 Litre of Tears.
Last Cinderella- has the same main actor as Boku no ita jikan.
I'm not sure if you like cop shows, but Bitter Blood is pretty good. It has a bit of everything, comedy, romance, and action.
J-dramaland e youkoso. :)

A LOT of older dramas are masterpieces, but they're often quite tragic:

Kamisama mou Sukoshi Dake 
Koukou Kyoushi 2003
Majo no Jouken
Yasha
 Hikari to Tomo ni 

Newer, lighter:

Love Shuffle
Nobuta wo Produce
Legal High
Shikaotoko Aoniyoshi
11nin mo iru 
Stand Up
Beginners
Himitsu no Hanazono

Newer, darker:

Last Friends
Soredemo Ikite Yuku
N no Tame ni 
Kazoku Game
Kazokugari
Yakou Kanransha
LIFE

edit: Hmm... I guess Stand Up shouldn't be in the newer section. But whatever. It's good anyway.
For darker genres, try: Ouroboros, Alice no Toge, Guilty Akuma to Keiyakushita Onna, Liar Game 1&2, Bloody Monday 1&2, N no Tame ni, Ryusei no Kizuna
Family/friendship/slice of life themes: Ashita, Mama ga ina, Gakkou no Kaidan,Hayami-san to Yobareru Hi, IS - Otoko Demo Onna Demo Nai Sei, My Boss My Hero(must although is quite old), Tennou no Ryouriban, Wakamonotachi 2014,Yowakutemo Katemasu
Lighter/Romance genre: Atashinchi no Danshi, Buzzer beat, Kekkon Shinai, Legal high 1&2 (must), Love SHUFFLE (must watch for romance), MIss Pilot, Soratoubu Kouhoushitsu, Rich man poor woman, Saikou no Rikon,Taisetsu na Koto wa Subete Kimi ga Oshiete Kureta,Watashi ga Renai Dekinai Riyuu
You already have many recommendations here, but I will add my list as well. (Maybe it will make it easier for you when you see titles recommended by more than one person. And the "recommend me something" question is so hard to resist!)

Love Shuffle http://mydramalist.com/115-love-shuffle (friendship, comedy, romance) It’s a really good one!

Orange Days http://mydramalist.com/264-orange-days (friendship, romance) Not comedy but a great, warm story, the actors have chemistry! If you hang around a little longer, more people will recommend it for sure!

Saigo kara nibanme no koi http://mydramalist.com/3156-saigo-kara-nibanme-no-... (friendship, family, romance) This is a very fun, lighter drama, similar to Date in the way that the main characters don’t get along very well at first.

Slow Dance http://mydramalist.com/987-slow-dance (friendship, romance) If you like Orange Days, you will like this one, too.

Last Cinderella http://mydramalist.com/6527-last-cinderella (romance) I know some people don’t like it but I really enjoyed it!

Amachan http://mydramalist.com/6182-amachan It’s a long drama but each episode is only 15 minutes! It has a little bit of everything, almost!

Other good dramas that are not exactly what you are asking for but worth checking out anyway:

Bambino! http://mydramalist.com/126-bambino (life, friendship, food)

Biblia koshodou no jiken techou http://mydramalist.com/5996-biblia-koshodou-no-jik... (friendship, investigation, books!)

Koukou kyoushi 2003 http://mydramalist.com/790-kou-kou-kyoushi-2003 because it’s great and I always recommend it to everyone!

Thank you all for the recommendation. It should keep me busy for a while :)

Some of these I've already checked the synopsis before but wasn't sure if they were any good but after reading your endorsements I shall give them a try.