I wholeheartedly agree with all of Curiosity's suggestions. I would like to add to this growing list of anime goodness:
Ninja Scroll - an excellent piece, definitely a classic. 'He has fallen victim to the throat needle' is still one of my favorite anime lines to date...
Spriggan - also pretty good, a little off-center.
Outlaw Star - excellent series, a little old (coming up on it's 10 year mark) but worth it.
12 Kingdoms - a totally under looked series with an amazing albeit a bit complicated storyline. The characters are plentiful but well thought out and developed throughout the entire story. It's lengthy (50 episodes) but totally worth it. It also happens to have really cool music. This was originally a manga, but it was sadly never translated to English (as far as I know). Even though there are very few sub plots that are left open (which happens more times than naught in Anime - see below), this series delivers an epic storyline with rich character development.
Berserk - great series, open ended. It was a manga first, and the Dreamcast Game 'Sword of the Berserker' actually reveals more story and picks up where the series left off.
Honestly, Akira was great and all, but it didn't really blow my skirt up. It's pretty dated, and there are several other peices out there that are simply more enjoyable to watch. My intro to anime was a Fist of the North Star / Neon Genesis Evangelion one-two punch. I highly recommend both, the latter more so than the former. Fist was good because it was completely over the top and required total suspension of disbelief through most parts, even for anime. You'll know what I mean after you watch the scene that involves a building falling on the main character. That may seem like an 'Oh shit' moment, but he simply keeps walking, said building barely scuffing his Pumas.
One of the major problems I have with anime is that some pieces simply end with no resolution whatsoever. Take Blood - The Last Vampire. It introduces enigmatic characters, and promises riches in the form of a tantalizing story and interesting characters. It tickles the scrotum region, simulating feelings of trembling anticipation for what lies beyond the initial scenes. Like a fine apartif before a meal.
And then it's over. Credits.
You've been invited to dinner, sweet talked with the potential of an involved plot and deep character development. And just when the small talk ends and the conversation begins, you look up and realize you've been stiffed. You're now left unsatisfied by a lackluster meal and also left holding the check. Examples of this phenomenon are The Heroic Legend of Arislan and RG Veda. Stay away.
Much like America television as a whole, there's is a CrapTone of anime out there. But a lot of it is bad. Like whiny Episode 4 Luke Skywalker bad (who probably would have gotten bitch slapped and robbed by a male prostitute if he was allowed to waste time with his friends at Tosche Station, but that's a different story for a different time). Some are even creepy. Like the whole Traveller / Westley Crusher sexual deviance thing ('What a nice, tight, uniform Mr. Crusher... <drool>'). So the only real way to find those gems is to go balls to the wall and just start watching everything, bracing yourself for the steaming nuggets of shit that you're undoubted going to stumble upon.
That's my $.02
He who fights with monsters might take care lest he thereby become a monster. And if you gaze for long into an abyss, the abyss gazes also into you...