I recently came upon a rather impressive anime, called * Berserk*, whilst browsing my local video store. It stars this guy with a sword as big as he is, who joins a mercenary league known as “The Hawks”. With each passing episode, the leader of “The Hawks” seems that he will become evil at some point in the story, but I haven’t gotten that far yet.
Another vote for Ninja Scroll here. Great introduction to the genre.
EVANGELION: NEON GENESIS is quite good. The sexual undertones are never actually brought to the surface, though.
If you do watch Berserk, just don’t expect the first episode to really have anything to do with the rest of the series. 
(Note: I didn’t finish the series either.)
If you really like cars and street racing, watch Initial D 
It’s probably my favorite anime other then Chobits and Trigun.
If I hear one more person talk about how good Cowboy Bebop is I’m gonna snap!
I grant you it was ok, but it is not the END ALL BE ALL of anime. To be honest I found it a bit pretencous at times. And anyone who says Cowboy Bebop is a space drama is smoking crack. Cowboy Bebop is an old fashioned “Gumshoe” type drama thrown in space to give it a SF feel. That’s it.
Lets see, first time watcher Anime?
Tenchi Muyo - I can’t believe no one mentioned this. This is about the best anime in terms of character development that I have ever seen. Very dramatic at times, but tends to lean to the sit-com side every so often.
El-Hazard - Same as Tenchi, only without the great character depth and starting the ever popular Mr. Fujisawa!
Escaflowne - Very cool if you’re even remotely into fantasy. Probably one of the best written stories of any anime I have ever seen. There are no real “bad guys”. Just people driven by their own motives. Very detailed.
Hellsing - Modern day vampire hunting series. Very artistic. It’s what Cowboy Bebop should have been in my opinion in terms of atmosphere.
Bubblegum Crisis 2040 - Think “Ghost in the Shell” the series. But with better characters in my opinion. Very dramatic at times, and packed with action at others.
If you prefer movies to OAV’s (Originally Animated Videos) I recommend the following:
Akira - THE anime. This is what really kick started the genre in the US. Very serious. Very dark.
Ghost in the Shell - As was said before, what the Matrix WISHES it could have been.
Patlabor 1 & 2 - For the same reasons as listed above.
Ususei Yatsura, known in the West as Lum.
Boy meets girl. Boy lusts after girl (also other girls). Earth is invaded. Boy meets girl alien. Boy lusts after girl alien (also other girls). Girl Alien gets idea that she & Boy are engaged, she moves in with him. And with his parents. Invasion called off, boy stops lusting after alien girl, still lusts after Earth girl (also other girls). Wackiness ensues. Very funny.
I third (fourth? fifth?) Kenshin, its a nice not too wacky anime that has its hillarious moments as well as decent stories (my gateway drug) watch the two OVAs after you are hooked on the series. Oh for the record, I was hooked from the moment when Kenshin made his litte sword speech in the first episode.
And another one for Cowboy Bebop, its just a must see (especially the movie, Knocking on Heavens Door).
I liked trigun up to the last few eps, why do the makers of decent anime so often have the urge to shove some pseudo philosophical bullshit down our throats?
Maybe not a gateway drug but Vampire Hunter D Bloodlust is a must see also, maybe one of the prettiest artworks out there.
Ninja Scroll was ok, but didnt do much for me when I saw it, plus the dubbing is gruesome.
Never get any dubbed version of any foreign film, anime or otherwise. The subtitled version is always better.
I was going to recommend Jubei Nimpocho (also known as Ninja Scroll, but a bunch of folks already beat me to that one. So I’ll second Tank Police instead. Sort of science fiction; the police start using tanks, because the mayor won’t let them use tactical nuclear weapons.
And unless you really like weirdness, pass on Akira. There’s a slim chance that if you’re suffiently stoned/drunk/sleep-deprived that you’ll sort of vaguely understand it, but it’s not likely.
Well, first you need to pick up all 6 DVD’s of Cowboy Bebop and watch those (and enjoy the sound of Skippman snapping).
Next, get the Bebop movie, Knockin’ on Heaven’s Door. It’s set between episodes 24 and 25 of the series, so watch the series again, then the movie, and then the last two episodes.
Next, check out Castle of Cagliostro. It’s an action/slapstick movie about a master thief Lupin III (a well-known manga character in Japan) and his sidekicks. It’s really more an “Intro to Studio Ghibli,” though, since it doesn’t have the same tone or style as the other Lupin stories (I’m told; I’ve only seen one of the other ones and I hated it).
I’d say definitely give Serial Experiments Lain a shot. It’s unlike any other anime series I’ve seen; I’d compare it more to “Twin Peaks” than any other anime. In brief, it’s about a near-present-day school girl who finds herself getting drawn more and more into the internet.
Try a couple episodes of Escaflowne. A lot of people are turned off initially, because it starts off very “girly.” It’s about another present-day school girl, except she gets pulled into another world where a war is being fought with airships and giant dragon robots. There’s a lot of soap opera to it, but also some fantastic action scenes and a pretty involving plot. I don’t recommend the Escaflowne movie, though, since it tries to cut a 26-episode TV series down to 1 1/2 hours; it loses a lot, obviously, and feels rushed and kind of incoherent. Sunrise, the company that made Escaflowne, also made Cowboy Bebop, so you should watch Bebop again afterwards.
The Gundam series, as well as Robotech/Macross, are widely available. I was never able to get into either, but YMMV. What I’ve seen of G-Gundam makes me think that it’s just wacky enough to be appealing. I grew up watching Starblazers and Battle of the Planets, which are both available on DVD now, but I can only recommend those for nostalgia value.
You should see if you can find some more Studio Ghibli movies – Cagliostro, My Neighbor Totoro, and Princess Monoke are the only ones easily available in the US, and they’re all good but not the best, IMO. The best would be Porco Rosso and Nausicaa of the Valley of Wind. (Actually, my favorite is Pom Poko, but I seem to be alone in that.) You can find synopses of all these at Nausicaa.net. I managed to find a, err, “unofficial” set of all the Ghibli movies, you might be able to find something similar. (Hint: Ebay). While you’re looking for those, watch Cowboy Bebop again.
A couple of other series make reference to other anime, so I think you can enjoy them more after you’ve seen the older stuff. Neon Genesis Evangelion is one – don’t be scared off by the people who tell you how weird or disturbing it is; it is pretty dark, but it’s good stuff. Another is The Big-O, which is a very cool-looking series with giant robots duking it out in an art-deco city, but be forewarned that it was never actually finished. And yet another series with anime references is Cowboy Bebop, so go ahead and watch that again. Notice that Spike Spiegel has Lupin feet and that Jet Black looks a little like Goemon and Inspector Detector from Speed Racer.
Finally, if you’ve still got Cartoon Network be aware that they have an “Adult Swim Action” block on Saturday nights, at least in the US. There’s some great stuff on there, like Yu Yu Hakusho and Cowboy Bebop, and then some lousy stuff like Pilot Candidate.
And my favorite anime website is Anime Jump! They don’t get too bogged down in all the lingo that most anime websites use, and they don’t assume that you’ve already seen everything at anime conventions; if you know what Akira is and can recognize the words hentai and otaku, you’ll be fine. Their reviews are reliable. Especially their review of Cowboy Bebop, which they correctly give an A+.
Fist of the North Star! Ow stop hitting me! You’re probably going to want to avoid it, given it’s so bloody, but it has some killer lines. " You fool, you’re already dead. You just don’t know it yet!" I liked it a lot more than the episode of Cowboy Bebop I saw but that’s just me.
I haven’t seen a lot of anime, at least compared to people like my best friend, but I liked things like * Princess Monoke, Wicked City* and Akira.
YYH being great except for the fact that the dub ranks quite high on my “makes my ears bleed” meter, behind only the Ranma, Fushigi Yugi, and Samurai Shodown movie dub.