So you want a romance story where there is some drama? Secrets, surprises, tears, rivals?
Angel Beats - Guy appears at high school and disturbs the lady with the sniper rifle.
Clannad - High school romance. Classic.
Full Metal Panic! - He's assigned to protect her but has no protection from her.
Mahoromatic - Killer android with tragic backstory is a maid and wants bigger boobies. Boobies warning.
Please Teacher! - You're married to your alien teacher. No problem, obviously.
Angelic Layer - A girl learns to play with fighting dolls. Romance thread in background. Set in same universe as Chobits.
Ah! My Goddess - A guy is offered a wish by a goddess. 2 seasons on http://hulu.com/ (first episode slow due to character development)
My Bride is a Mermaid - You'll know it's not a "The Little Mermaid" retelling when the yakuza show up, with the zany nearby.
The Melancholy of Haruhi Suzumiya - High school girl is a god. Subtle romance.
Kamichu! - Middle school girl becomes a god. As much romance as there should be in middle school.
Planetes - Space junk collectors. Realistic space travel.
Inifinite Stratos - School for mecha suit operators is full of girls, and one guy. Harem with some ecchi.
Shuffle! - Fellow discovers he has two fiancees. Good example of a harem show.
Maid-Sama! - Girl is important but has difficulties in mostly-boys school.
What is good really depends upon what you like. Explore and discover what it is that you like.
There are a number of places with assorted anime. Crunchyroll.com features subtitled stuff, including some brand new shows. AnimeNewsNetwork.com points out more things. But I suggest you start with Hulu.com because of there are popular dubbed shows there (although it's getting crowded with 4-episode samplers).
Go over to Hulu.com and look in Genre > Anime. You'll find an assortment of stuff.
First, register for a free Hulu account. This will let you add things which might be of interest to a Queue (list).
Then go browsing. Try an episode or two of things which seem slightly interesting. Keep in mind that sometimes the first episode is different than the rest of the show ("Ah! My Goddess" focuses on one character in the first episode, "The Melancholy of Haruhi Suzumiya" shows a student movie, "My Bride is a Mermaid" starts looking like a "The Little Mermaid" remake until the yakuza show up just ahead of the zany).
If you find something that you like, look it up on places like the http://AnimeNetworkNews.com/ encyclopedia, so you learn how to describe what you like as magical girl, fantasy, or mecha. Places like http://anime-planet.com/ have recommendations for similar shows.
You'll probably find at least a half-dozen things of interest on Hulu.com and you can explore further with related things. Some shows may seem kind of interesting, such as maybe a romantic comedy (romcom) where the characters or situation don't quite appeal to you, but that might lead you to explore similar shows until you find a mix which is more interesting to you.
