I'll suggest the ones I started with (besides Sailor Moon/Pokemon/Cowboy Bebop that they used to show in Toonami or Saturday Morning cartoons, that I didn't even know were Japanese cartoons lol)
The first time I voluntarily watched anime was for Fruits Basket - it's my favorite to this day, and it's romantic/comedy about an orphan girl that moves into a house with boys that turn into animals when embraced by the opposite gender (there are 12 people after the 12 Zodiac animals) - it's really a touching story http://myanimelist.net/anime/120/Fruits_Basket
and Ouran High School Host Club - a gender bender about a girl that is mistaken for a boy and forced to join the Host club to repay a debt (H.I.L.A.R.I.O.U.S) - they made a drama out of this too, but I haven't watched it http://myanimelist.net/anime/853/Ouran_Koukou_Host_Club
Soul Eater - ...it's hard to explain...it's got Death the Kid in it, that should be reason enough (: http://myanimelist.net/anime/3588/Soul_Eater
(I only found the English dub, I liked it better in Japanese)
I haven't watched anime in 2 or 3 years, so they may have come up with better stuff but the old stuff is always the good stuff, right ;)
I'm pretty picky when it comes to Anime as well and it wasn't easy for them to catch my eye.
I really enjoyed Clannad and Clannad After Story, it was a really great series, has a very slight bit of supernatural in it but not much, otherwise just a slice of life romcom.
I also second Kimi ni Todoke, it's very cute and innocent, and it even has a Japanese live action movie as well.
Eden of the East is a very well done and very unique story that I would recommend as well, it's definitely not your norm when it comes to anime watching.
Chobits is a classic, very funny but it is more on the mature side of comedy, and it's a science fiction romcom.
I'm also very picky with anime!!
My preferences are limited to:
action/psychologicall thrillers like Death Note, Steins;Gate, Darker than Black, Monster,
slice of life like AnoHana, Usagi drop, Welcome to the NHK, Hyouka
romance like Nana, Toradora, kaichou wa maid sama, Lovely complex, Kimi ni todoke, Nodame Cantabile...
Evony wrote: Some people recommend Gintama - I wouldn't. It's totally awesome (the best of the best imo), but not for somebody who's new to anime. It's episodic and the humour is really... abstract, plus many scenes are references to other animes, so you might be a little lost.
I agree this one. there are lots of things ( japanese history, other anime parody, ... ) in Gintama new anime follower mustn't watch.
if you new anime follower and want to comedi -romance you must watch Ouran High School Host Club and Lovely Complex after skip Beat,Nana ,Itazura Na Kiss, Paradise Kiss, Nodame Cantabile and toradora :)
and Baccono , Samurai Champloo, Ao no exorcist , Durarara , D.gray man you must show if you like action :)
Lovely complex also has a live action drama version, a movie. As well as Paradise Kiss, Kimi no todoke and Itazura na kiss (Drama: It started with a kiss and Mischievous Kiss)
For romantic comedy types of anime, I also think that Kaichou wa maid sama, Ouran High school host club and Kimi no todoke are great choices. I also liked Special A, it has slow romance though, same with Ouran. Fruits basket is good too but I prefer the manga, since it has a great ending :D Skip beat also have slow romance but with a good plot. (Btw, skip beat also have drama, I haven't watched it yet though) Also, marmalade boy has a drama, the anime is really long though.
Tonari no Kaibutsu-kun and Kamisama Hajimemashita are great animes too. (KamiHaji is fantasy but I really liked it)
Since you have a bunch of good recs i'll just give you a few i think a drama lover would like.
Natsuyuki Rendezvous very drama like
Rec. This is a short anime
Sukitte Ii na yo probably my top pick and once you are initiated you'll probably enjoy kotoura-san
What about kimi ni todoke.
Hope you eenjoy. I went the other way (anime first) and love the Japanese language so our taste might be very different but hope it helps
If you like romance and comedy..
My first anime was Chobits, very cute and funny ^^
Bokura Ga Ita - it's seriuos romance, but very beautiful, there're 2 japanese live action movies also
Clannad - one of my fav animes, it's about friendship and love, a bit of fantasy also. Serious and touching *sniff*
Lovely Complex - sweet romantic comedy, very funny ^^
Onegai Teacher - student-teacher relationship :>
REC - about work at radio station :)
Bokura ga Ita: Serious romantic series, but great to watch
Death Note: Not a shoujo (girly girly) but great... just great and little bit psycho :3
Itazura na Kiss: Maybe you saw the drama versions (Playful kiss, It started with a kiss etc.) lovely story c:
Kaichou wa Maid-sama: Great chemistry and hot USUI! <3
Kimi ni Todoke: Cute love story :3 (Maybe saw the jap. movie)
Skip Beat: Sooo funny and hot guys *¬* (here also xd maybe saw the chinese drama version of it)
Ouran Koukou Host club: Very funny, not really romance but hot guys <3
Special A: Kinda reminds me of Kaichou wa Maid-sama <;
Sukitte Ii na yo: Really cute love story, but manga is better, anime is waaay too short kekeke :3
Yamato Nadeshiko Shichi Henge: not really much romance in it, but it's cutee tehee~ (also here, maybe saw the drama version of it)
That's my list for you, hope you get a great watching time~
I would suggest things like...
Ouran High School Host Club
Kuragehime
Kaichou wa maid sama
Kimi ni Todoke
Fullmetal Alchemist: Brotherhood - it's not a romance comedy but it's my favorite!
