I think it's unfair to call elfen lied shit. It wasn't really my thing, but the story really wasn't that bad either.

As for my suggestions:
Bakmonogatari - boy meets weightless girl and the mysterious events start rolling
(Neon Genesis) Evangelion - it ought to be seen at least once whether you like mecha or not
FLCL - .....just watch it, I can't really explain this one and it's only 6 episodes
Cowboy Bebop - not my favorite, but worth checking out
Shakugan no Shana - overall one of my favorite series
Ouran High School Host Club - technically a girls' show, but it's just so funny that it doesn't really matter
Mushi-shi - not a lot of action, but it has good artwork and music. Set up more or less as a series of short-stories and each episode/story is pretty good story-wise
Pandora Hearts - might be a little cliche, but I think the series still pulls it together pretty well
The Third: Girl with the Blue Eye - if you can find it, I'd suggest checking it out. Not sure what traits to emphasize, but I thought it was pretty good
The Melancholy of Haruhi Suzumiya - some like it, some don't, I do, and I highly suggest it
Lucky Star - if you like moe, there's a good chance you'll like it, if you don't... then try another series. Based on a 4-panel manga it's random hilarity that I find quite enjoyable
Familiar of Zero - fantasy series. English dub isn't the best, but overall it's a pretty good series.

Notes and disclaimers:
1. My avatar is from an omake of Shakugan no Shana called Shakugan no Shana-tan. The character is "Shana-tan", a chibi version of the character Shana from the series.
2. I watch all my anime dubbed if a dub is available, but I've also seen subs for pretty much all of them too (criticism is not appreciated)
3. I strongly prefer happy endings, so most of the ones I suggest fall into such categories.
4. Most anime I suggest is licensed, so keep that in mind
5. All suggestions are my personal opinions, you are free to agree or disagree with them, but please be polite

That is all.