Okay so I never watched a Japanese drama before. I only watch K-drama, so I decided to watch Japanese for the first time!
What are some good J-dramas? I like mystery-romance, so if you ever watched this kind of drama and you liked it, please write its name.
Thank you ^_^
hard nut! (its mystery/ crime solving which was a fun watch)
otenki onee-san (this one is really unique. its about a weather genius who helps solve crimes)
doctor X (about a genius doctor who is just an overall badass. she's awesome)
mop girl (bit of time travel combined with murder mystery.)

those are some of my favorites =)
Mystery-Romance:

Guilty - Akuma to Keiyakushita Onna
Koori no Sekai (Ice World)
Sora kara Furu Ichioku no Hoshi (One Million Stars Falling from the Sky)
Byakuyako

Romance:

Love Shuffle
Majo no Jouken
Kou Kou Kyoushi 2003
Date

Mystery:

Kazokugari  
Kazoku Game
N no Tame ni  
Yasha
Alice no toge - has and element of mystery to it.
Ishi no Mayu
Although I wouldn't necessarily recommend it as the best starter Jdrama, N no Tame ni probably fits the mystery/crime/romance genres best.