-It's not what I would call very Japanese, but I'd recommend Cowboy Bebop to anyone, anime fan or not.
-Neon Genesis Evangelion is essential anime, but I wouldn't guarantee that you (or anyone) would like it.
-I'm about 10 episodes into Samurai Champloo and I'm loving it so far.
-I'm following Monster as it airs on the Sci Fi channel. The animation leaves a bit to be desired, but the writing is great.
-Death Note is really good, although I don't think I'm that likely to watch through it more than once.
-Tengen Toppa Gurren Lagann has good comedy and action. Its plot has some minor problems but it's good overall.
-The Lupin III movie The Castle of Cagliostro is great. I haven't seen any of the other ones yet.
And since this is a recommendation thread, I have two questions:
1. The next anime box set I want to get is either Gungrave or Outlaw Star. Which is better? I'll watch episodes before deciding, but I'd like to see others' opinions on them.
2. Is Rurouni Kenshin good? I generally don't get into shonen anime/manga much, but I always hear it's good and it doesn't seem to have any haters, either.
I preferred Outlaw Star, which sorta seems to follow a shonen theme, though maybe it's more seinen, so you'll probably enjoy it.
As for Rurouni Kenshin, it was actually very good. If you enjoy that sort of thing. It doesn't follow a normal shonen approach in terms of preachy drawn out speeches from emo 14 year olds to hardheaded 30 year olds. The action is really good, and doesn't spare on any front, nor drag on with any issue for too long.
Also, I would recommend it's OVA Samurai X.
Evangelion is good if you can withstand emo levels of characterization on a constant basis. While most attribute it to being a mindf***, I only found that it was really that way in the last episode, when you were trying to decipher what it meant. Otherwise, it was a nice little drama (emphasis on melodrama at some points) which backed it up with nice aesthetics, planning, action, and mystery. I found the twists towards the end were a little obvious, but it was all in all a definitely good watch. The box set was well worth the money I put into it. And, as the credits constantly reminded you, it had "plenty of fan service" Also, and this just may be biased, I liked "Fly me to the Moon" rendition by Claire, because we share a name, and that's one of my favourite songs by Sinatra ever.
Otherwise, I give it the Françoise "must see" recommendation.
It also really depends if you like really short works, or really long lasting works as well.
Detective Conan (Case Closed) is also one I would recommend, personally. But then again, I like detectives. It's a story about a kid Jimmy (Shinichi) Kudo who's a high school detective, and best friends to Rachel Moore (Ran Mouri). He gets transformed into a kid, and is forced to live with an unsuspecting Ran as he basically solves crimes for her private eye father without revealing his true identity. Romance and nearly impossible situations ensue.
The Big O was another series I would recommend. It was a backdrop of a negotiator in a city with no memories, an android as his Girl Friday, and he has a gigantic robot named "Big O" which he calls on to fight other big robot. Fun ensues. It's definitely got that Film Noir feel to it, so don't be suprised when you hear that smooth slow jazz in the background.
Also, pretty much any Gundam but Wing I'd recommend. But only cause I hate Gundam Wing.
And Gundam 00, I disliked that one, as well.