This probably isn't much help for modern anime tastes or for series (outside Eva) but my gateway stuff was the 80s and 90s manime stuff that came through Manga Entertainment or Streamline, all the usual hits like Akira, Ninja Scroll, Demon City Shinjuku, Vampire Hunter D (original film). Then for higher quality fare of course Miyazaki, along with things like Wings of Honneamise, Perfect Blue (anything Satoshi Kon really), Grave of the Fireflies. Like I said, very 80s and 90s :p
For a recent series Era turned me onto Mob Psycho 100 is probably the best thing I've seen, also Devilman Crybaby is all kinds of delightful insanity if you're into the dark, ultraviolent stuff.
Oh yeah and for a recent film example I really enjoyed was Your Name, great flick.