I tend to like the less weird and less hyper series. I prefer subtitles, I blame growing up in :finland:.
My partial list, off the top of my head:
-Anything by Miyazaki, ever.
-The Ghost in the Shell franchise.
-Guardian of the Spirit (?????? Seirei no Moribito)
-Acquired taste, but the Crest of the Stars / Banner of the Stars series.
-Black Lagoon
-Planetes
-Haibane Renmei
-Mushishi
Fun, but maybe less good:
-Darker than Black
-Full Metal Panic, which for some reason I find really funny.
-Samurai Champloo
-Baccano
There are plenty of titles that have been very influential. Akira, Neon Genesis Evangelion, Death Note, Fullmetal Alchemist, Tenchi Muyo, Elfen Lied, etc. which I mostly left off the list because I either didn't like the way they ended, didn't like them in general, find them now quite dated, or found them to be pretty extreme in violence or other aspects to make a general recommendation. There are also tons of anime which I haven't watched, and may be the bestest thing ever. And a ton I just can't think of right now.