My personal favourites are Death Note, Code Geass, Hellsing Ultimate, and Vampire Hunter D. I don't like most mechas I've seen (though I do like Code Geass and Evangelion), and I hate harems, as well as excessive eichi, and Naruto. I like sci-fis and fantasies.
Anonymous
Favorite Answer
Tokyo Majin is my favorite ever, you'll like it.
Chrono Crusade is my next favorite, it's getting kind of outdated now though.
El Cazador de la Bruja
Eden of the East
Shakugan no Shana
Kaze no Stigma
Fullmetal Alchemist
Fullmetal Alchemist Brotherhood
Kurokami
Shikabane Hime
Talyn
Aww. you don't like Naruto? D:
try:
Soul Eater
Blood+
Fate/Stay Night
Rurouni Kenshin
Claymore
Gun x Sword
Yu Yu Hakusho
One Piece
Bleach
Fullmetal Alchemist: Brotherhood
A Certain Scientific Railgun
Beelzebub
Angel Beats!
Basilisk
Black Lagoon
Busou Renkin
Corpse Princess
Deadman Wonderland
Fairy Tail
Hunter x Hunter
Kekkaishi
ScottieWoof
Angel Beats - Guy appears at high school and disturbs the lady with the sniper rifle.
Ghost in the Shell - Android cop.
Full Metal Panic! - He's assigned to protect her but has no protection from her. Mecha, but not of galactic import as some others.
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.
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.
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full metal alchemist
full metal alchemist brotherhood
soul eater
gintama
mahou shoujo lyrical nanoha A
one piece
fairy tail
beelzebub
nurarihyon no mago
deadman wonderland
hunter x hunter
infinite stratos
zero no tsukaima
shakugan no shana series
sola
ranma 1/2
seikon no qwaser
mai-hime
P1mpin Curly
yu yu hakusho
hajime no ippo