(Oh dear this is gonna be a long one)
Here's a list of notable shows I've seen and would reccomend others check out at some point or another. (In no particular order)
Fullmetal Alchemist 2003
Much darker than brotherhood and has a fairly solid anime original ending.
Fullmetal Alchemist Brotherhood
Mmmmmm. This show is such a feelgood show, best when binged. Even though it's not dark or super complex and all, it has a superbly crafted world and narrative. You fall in love with the characters and it's just increadibly satisfying. I'd say this is my personal favorite anime, just out of sheer enjoyability.
Has a big dip in the middle (to the degree I put the show down and didn't finish it for another year), overrated to some degree but still a very fun show. Not the most intellectual thing ever so if you demand your shows be 100% serious, not your cup of tea I'd guess.
Neon Genesis Evangelion
Despite the memes about edginess and the like, I can say it's a good series. It's far from perfect and despite what some may say not everything has a deeper meaning, although a fair bit does. I'd reccomend taking things in from each character's perspective for this one rather than strictly your own. If nothing else the general direction and fight choreography is very good. This one is definately not for everyone out there.
The Melancholy of Haruhi Suzumia
I'm actually in the process of watching this one at the monent on reccomendation of a friend; I've heard good things and bad things. It's far from perfect but speaks more as a show about the concept of escapism and entertainment than it may let on. It often giving a more 'realistic' take on typical tropes despite the tropes still being outside the general happenings of reality. It's a fun show although I wish it'd explore it's characters a bit more, although I hear that happens in the movie that currently acts as the series conclusion.
Gundam Unicorn (OVA series)
my personal opinion is that the TV adaptation absolutely butchers the pacing on this one so I'll give that one a pass. The OVA's however... mmmm. This was actually my introduction piece to the gundam franchise. Being such there was a fair bit I didn't understand at the time, yet still it was a pretty good all around experience. Enough so that I've watched it three times to completion... seven OVA's. Each time I had learned a bit more about the general happenings and what was going on. It's a good watch and some what of a modern reintroduction of the UC timeline as well as being it's 'conclusion'. The protagonist may not be well liked by some due to his general desire for passivity rather than conflict, however.
Studio Ghibli Films
Should speak for themselves, but if you aren't aware of these I'd give a few a check. I haven't seen all of them but from what I have seen they are generally exceptionally animated for their time and all around good. Think of them as kind of the japanese disney, if disney only released stories for a more mature audience.
Edit: okay I just realized I basically already posted this, aside from one show, but I consider this one to be the better post so I'm gonna flag the original.