Anime :
Another recommendation for Full Metal Alchemist. Very good series. The movie "Conquer of Shamballa" is merely okay though.
Project A-ko : One of the funniest pieces of anime ever made and it just gets better the more anime you know. Only watch the original movie. The follow on OVAs basically all suck.
Ghost in the Shell Stand Alone Complex and GitS-SAC II are both first rate series with some very thought provoking looks at some of the implications of technology. Plus Major Kusanagi kicks ass. The movies, both the original and Solid State Society are so-so, IMHO.
Black Magic M-66 : Another terrific piece of anime from Shirow Masamune. The scene where the two bots tear into a special forces group is one of the best combat sequences ever. Older piece of anime, but worth tracking down.
Appleseed & Appleseed : Ex Machina - The recent CGI movies are spectacular pieces of animation. There is an older 80's animated movie which is pretty awful.
Tank Police : Dominion - Just the first two OVAs from the 80's. The rest of the anime installments are average at best, but the first two are hysterically funny and have a really great soundtrack.
The Mangas for GitS and Appleseed are also very good, though very different from the anime for the most part.
Super Dimention Fortress Macross - The classic series that got brought over as part of Robotech. While 95% of the plot is there in Robotech, there is an ADV release/translation/spiffed up version that is probably preferable. The duel between Max and Myria is still one of the greatest mecha battles ever made. There's a gorgeously animated movie from '84, but AFAIK there has only ever been truly hideously badly dubbed version ever released in the US.
Death Note : If there was ever a series that embodied the
Xanatos Gambit this is it. Avoid spoilers like the plague though. The machinations of Light Yagami have to be seen to be believed.
Area 88 : One of the original OVA series, still very good. Very melancholy though.
The Irresponsible Captain Tylor TV Series: Not quite Inspector Clouseau meets Star Trek, but that gets a lot of what the series is like. Extremely funny. Later OVA series average at best.
Hellsing : There's two series the original OVA and the later Ultimate OVA series. The later follows the manga plot much more closely and is gorgeously animated, but the original series is still quite entertaining and has a first rate soundtrack.
The manga is also very good.
Vampire Hunter D : Bloodlust - Gorgeous animation, compelling plot, interesting world.
Dirty Pair Movie AKA Project Eden - The Lovely Angels movie, wacky mad scientist, terrific soundtrack and one of the best climaxes ever in anime. The "The Conspiracy of Flight 005" is also very good. The "Original Dirty Pair OVA Series" is pretty good. Avoid the later "Dirty Pair Flash" series like the plague.
Some of the american Dirty Pair Manga series by Adam Warren are very good. "A Plague of Angels", "Fatal But not Serious" and "Run from the Future" are the best and "Sim Hell" is decent.
The manga for Nausicaa is MUCH better than the anime, as good as many people think that is.
Big O - Is an interesting Film Noir series with Giant Robots tossed in. Never quite lives up to it's promise, but it's an interesting ride.