Apart from all the classics, I have two tv series to recommend. Unfortunately I can't find it anywhere (DVD/Blue Ray, download, whatever) so I don't know where you could find it, but if you do, it's fun.
The first one is called Thundersub. The story is of an alien force having invaded and captured Earth. The last and only hope, Thundersub, a special ship/sub/spaceship. It can deal with the enemy mothership, but it can't go to space before some modifications. For them they have to travel across the seas to get what they need. Not so easy when the aliens control the planet and having slaughtered much of the human population and the remnants are slaves and pets. That's not even the worst. The aliens are transforming the planet to fit their demands and time is running out.
A masterpiece of TV anime IMO.
The second one is about the West deploying a ICBM shield against Warsaw Pact threats. The commies don't like it and the WW3 starts. The main hero is a scientist.
Great action and story. Unfortunately I have no idea what the title is.

FullMetal Alchemist is fantastic. Trigun is really good. Ghost in The shell is good if you don't mind really complicated plots. Samurai Champloo is really cool. Afro Samurai, while short, has some really awesome fight scenes in it. Hayao Miyazaki movies are breathtaking.
That's pretty much all I can recommend.

Gundam UC universe.
Begin by watching the three movies set in the original MSG since the series gets to long at points. Then, watch the OAVs set around MSG like MS 08th Team, 0080 and 0083. Afterwards, you watch Zeta Gundam and then ZZ Gundam and finish it off with Char's Counterattack.
Other Mecha series I recommend is Patlabor and the Starship Troopers OAV.

If you like good television you should check out Planetes, it is a fantastic anime.

The Cartoon Network didn't exist back then, not even the US version.

did any one say Desert Punk yet

Mushishi
http://anidb.net/perl-bin/animedb.pl?show=anime&aid=3433

Spice and Wolf
http://anidb.net/perl-bin/animedb.pl?show=anime&aid=5406

And of course
Irresponsible Captain Tylor
http://myanimelist.net/anime/569/Irresponsible_Captain_Tylor

Theres also Kemono no Souja Erin
http://myanimelist.net/anime/5420/Kemono_no_Souja_Erin

Then again, its all subjective taste

The Twelve Kingdoms

http://en.wikipedia.org/wiki/The_Twelve_Kingdoms

Death Note was pretty good, though I preferred the 2 live action films as the anime went a bit OTT towards the end. Another 2 live action films I heavily recommend if you can find them are Crows Zero & Crows Zero 2 which were directed by Takashi Miike and based on a manga about a school where kids fight to be the best.
As others have said, Elfen Lied is a good one.

I also really enjoyed Blue Gender which is a series where the earth has been taken over by these giant bugs (starship trooper-esque) and mankind is living out in orbit on a ship and making incursions from time to time

Air Gear which sadly has a cliffhanger ending but was a good series based around these hyper powered rollerblades (reminds me of Jet set Radio.)

Wolf's Rain which is set in the future where people believe that wolves are extinct but they've actually mostly taken on a human form to disguise themselves and it follows a band of wolves seeking out the promised land.

These are all animes of around 26-32 episodes as I don't really like watching long running series with no end in sight because variety is the spice of life.

If you like cyber punk you could try AD Police, Cyber City Oedo 808.

Also worth checking out are Ergo Proxy, Ghost in the Shell, The Shinkai Collection, Noein... seriously, I could list stacks of them.

Damn, that was a bloody classic. Especially the Manga Entertainment version. The new soundtrack they added to it was better than the original (if you ask me).

It really depends on your age, what you are interested in, etc. I don't watch a lot, but I have some favorites:

Hellsing Ultimate - This is more faithful to the manga than the original series, and has quite a bit of humor in it, as well as shit tons of violence. Interestingly enough, the voice of Alucard in the English dub? Togusa from Stand Alone Complex.

Ghost in the Shell: Stand Alone Complex - Not like the movies, so if you are basing your opinion off of them, don't. They are high quality, with gripping storylines, lots of action, and a bit of humor thrown in at the right times.

Ranma 1/2 - Hilarious anime, and arguably the first real "Harem" anime. Incredibly silly but lots of fun.

Azumanga Daioh - One of my absolute favorites. Don't let the fact it has only schoolgirls in it, it's cute, funny, and has loads of memorable moments.

paprika

fav ghibli, POM POKO

I'm surprised that anyone recognised Cyber City, obscure anime these days, but still pretty amazing. I'm not sure which soundtrack my version has, but I did own the original VHS version and I think it's the same... that's not much to go by though, I thought the same about the Armitage III Poly Matrix, and it turned out that all of the voice actors were different :P