I personally like Deathnote and Cage Of Eden, especially cage of eden if you didnt read it yet I advise it to do so.
I have started learning Japanese just a month ago and tried reading a Manga in furigana and I realized Manga seems not a Good Idea for a beginner since its in too casual form. try NHK web easy though its too dull.
It is great !!!! I love Levi .
You should read it .
My favorite throughout high school and college was "NANA" by 矢沢 あい (Yazawa Ai), and pretty much anything by that author. Her artwork is amazing. I used to get really excited whenever I saw a new volume on display at the bookstore.
If you like horror, 伊藤 潤二 is the master.
For humorous, I suggest "ボンボン坂高校演劇部" and "浦安鉄筋家族."
For mystery, I know "名探偵コナン" is more popular, but I always liked "金田一少年の事件簿" better.
I stopped reading manga over ten years ago (so all my suggestions are pretty old) and these are the ones that left an impression on me.