Higurashi indeed is VERY awesome.
First season (24 eps or so) is very what on earth's going on and all that, may be interesting if you like that sort of thing, but then after that the second season gets very amazing.
Not just death note like making you think epicness (which is also very good), but also gets very emotionally awesome.
However, if you plan on watching it (highly recommended) then you need to watch the first season and can't just skip to the second though.
Will be very worth it in the end though.
As for some other stuff, there's probably plenty I'm forgetting and of course I won't be just naming any old crap I saw but stuff that I actually think is very well worth watching:
Comedy: School Rumble - No idea if it depends on the person, but I couldn't stop laughing and love it.
Romance: Clannad/Kanon/Air (by kyoto animation, not the short films by some other company) and ef (a tale of memories), possibly even Elfen Lied (if you're up for having your heart destroyed).
General: Full metal alchemist (still not gotten around to seeing the latest "remake", sure it's awesome, but the original one is VERY good), Fate/Stay Night is another favourite of mine and the same for Ghost in the Shell.
I'm actually watching GTO (drama version) on TV lately as well. Saw (most of >_<) the anime some time back and quite liked it.
Should try out a few eps to see if that sort of thing is to your liking too lol
That's all I can remember that I really recommend right this instant I guess. I'm sure there's more though, if you ever need more
I think I watched the first 4 or 5 episodes of Eureka 7 and was utterly bored. 
I remember thinking the same now that you say that.
Indeed the start is slow and just feels like some random anime, but after a bit it turns out to be very serious and dramatic.
I remember when I realised it was getting more serious and being very relieved that it wasn't just any old crap but actually going to have a pretty good story.
