I wanna watch a drama where the female falls in love with the male lead first.
Female lead likes or admire the male lead.
Female lead takes the initiative. 

Preferably Korean Or Chinese.

Examples:
"My Love From Another Star"
"Innocent Man"
 Like how Yuan Shan Shan fell in love with Wallace Huo in "Swordsman"
Did you watch Diamond Lover (Tang Yan, Luo Jin, Rain) and Sound of the Desert (Cecilia Liu, Hu Ge, Eddie Peng) yet?
Best Time (Wallace Chung and Janine Chang)
In Time With You (Ariel Lin and Bolin Chen) - only in the beginning though
Yun Zhong Ge (Angelababy, Lu Yi, Du Chun), I would say this fits it since Yun Ge has been taking the initiative to pursue her first love.

Agent X (Tiffany Tang and Luo Jin) - I can't remember who took the intiative, really, but good drama nonetheless.
If you watched all of these, sorry :(

On the other hand, there's Love Transplantation (Chen Xiang and Ying Er), which is a movie that started off with girl chasing boy.
If you don't mind Taiwanese, the best version of this is It Started With A Kiss, and the second season They Kiss Again. She is already in love with him at the first episode, and she lets him know in that episode. There is a Korean version of it called Playful Kiss, but it is the WORST version. There is a Chinese "Adaptation"Say That You Love Me, but it follows it very loosely, and I dropped it so I don't know how good it is.
I really liked Say That You Love Me so I can reccommend that one. I also disagree with @squeakyelle about Playful Kiss. :p While not my favorite version of Itazura na Kiss, I don't think it was the worst.

Others I have seen that fit your criteria:

Tomorrow's Cantabile
Pretty Man
Reply 1994
King Flower
Fifteen Years Migratory Birds, supposedly the female lead falls in love first tho I don't know if she ever takes any initiative or simply quietly pining after him. I like this trope too, will be keeping my eyes on this thread xD
Try the princess man....well the both feel in love to each other at first but it's the girl who always chase after him and support him in any ways
Hong Gil dong: the guy is very cold to her and ask her to not follow him but she always chase him and follow him everywhere.
You Are beautiful
Nice guy 
Bad guy
It's like this theme
I'm looking for dramas like that too... although its very very few of those and hard to find. It seems like this topic is really out of the "standard" in dramaland! I'm so sick tired of all those up-guarded female leads that allways are on fire. hope the trend'll changes soon....

Anyway, back to your question... Those I have watch so far that fits the qualification is:

It starts with a kiss (TW version of Itazura na Kiss)  – It has a very silly and funny female lead.

Kiss me (Thai version of Itazura na Kiss) – The female lead in here is not quite silly as in the TW one, but is more reasonable and very cute!

Nodame Cantabile (J-drama) – The female lead literally chase after the male lead. Crazy and cute female lead.

The Palace (C-movie from 2013) – Cute, quick and light story about a servant who falls for the prince.

The Mystic Nine (C-drama) – Although the romance is not the main part of the storyline, the romance development between the leads are cute! 

My girlfriend a Gomiho (K-drama) – Cute romance story between a human guy and a fox-beast lady

Perfect couple (C drama) – A light silly-funny wuxia romance drama with Wallace Huo. Not the best drama, but it fits the qualification you are looking for.

Prince of ling lang (C-drama) – Best of "modern-wuxia" drama I have watched. If you love Swordman, you'll love this drama too. The story is so fine!

Que Sera Sera 
(K-drama) – its about a fraudster guy falls for the new next door neigborhood girl.

Lucky Days (C-drama, I think or TW...?) – A bit off the topic but... I think you gonna like it. its about a marrid couple. This is kind of "back to the future" sci-fic  drama. where the husband goes literally back to the past to fix the relationship problems with his wife and learn more things about her. Tammy acts very cute and touching in this drama.

W  (K-Drama) – A sci-fic drama. A mystic action-romantic story between daugther of a comic author and the comic character drawed by the father. All sort of feeling will drag out of you when watching this drama. I love this drama very much for its creativity! A brillant drama!

The fox fairy court
 (C-drama) – A mixture between the originally foxfariry-human-romance fairytale and case-Investigations story. LOVE the chemistry between the main leads! And the main actress potrays her character in so adoarble way. She is so freaking cute in here!

Smile, you (K-drama) – A nice romance comedy drama about a rich girl who being dumped at her wedding day due to her family's bankrupt. Of sympathy due to their situation, Her family's driver takes them all to live together with his family. The girl then falls in love with the driver's grandson. And tries to make her way to catch the grandson's heart. Thats the start of developement of a cute and strong bond between them two.

Bring it on, ghost! (K-drama) – A sci-fic romance comedy drama. Its about a girl-ghost whos teams up with a college guy to fight agaist evil ghosts. In the process the girl fall for the guy. And her funny actions begins.

Decendants of the sun (K-drama) – Story of the secondary couple in the drama will fits the topic's criteria perfectly. Its about a cat-&-mouse run between daughter of a commandant and a master sergeant.

Hua Jian Ti Hu Fang Da Chu (C-drama) – I will category this drama as "modern-wuxia". its about a lonely girl who is good at cooking. And by accident she gets married off into a rich family, to a guy knows for his "rudeness". Instead  of running away, the girl falls in love with the guy quite quick, and trying to understand his rude behaviours and help him express him self. I like story and the harmony between the characters, but I wish for some more spice in the romance process.

Thats all I can think of for now.
Is there any other dramas like this??  See my list in previous post, above here.
The Princess' Man ---  although they both fall in love at same time ..... throughout entire drama, she is the one being pro-active in pursuing and preserving their love.

East of Eden ---  She falls in love first... she is the one who actively chases after him, never gives up on him ..... she's wonderfully smart, clever and brave !
Rich man poor woman 
another miss oh

I'm crave for more... more please more more more!!

 TeaBear:

I'm crave for more... more please more more more!!

I mean it's not romantic initiative but she disrupts his life and changes him... Fox's Summer.

The Wife's Lies has a girl who chases after the a guy the entire show pretty much. 

I wouldn't say either show is spectacular but the female leads are much more assertive/pushier than the men they are hoping to pair up with.

Sweet 18

Another oh haeyoung

Lovers in prague

Please love the useless me

Queen inhyuns man

Will it snow for christmas

oh my ghostess

My sunshine

Girls generation 1979

pasta