Just a note: Watch all of these (except RoboTech where it's not an option) in Japanese w/ English subtitles or don't bother! I used to think that was a bit purist but, after hearing the English dubs of some.... *cringes*
Cowboy Bebop
If you like Anime at all (if not, then why are
you here?), make it a priority to check out Cowboy Bebop.
And give it at least four episodes (it seems a little weird at
first). If you're not hooked by
then, it's not for you (I suspect you'll be hooked). Between
the artwork, the quirky characters and character development and
the jazzy music by Yoko Kanno (even the
CorpsVets senior drum corps
thought this stuff was awesome and use it in their show!), this is
one of the most compelling series with overall appeal (and a lot
of fun too!).
Robotech
A tried and true favorite from my college days.
I thought sure this was going to be a "Transformers" thing when I
was finally pressured into watching it... Boy was I wrong!
I was glued to the TV for days as I played the video series though
all 85 episodes! There are a few characters I'd like to shoot ("Minmei
hunting, anyone? WHY do you people love her?!") but it was the
most 'adult' cartoon that passed into the kidfare anime genre I'd
ever seen at the time.
Record of Lodoss Wars
Cross anime with Dungeons and Dragons and you've
got Lodoss Wars.
The Vision of Escaflowne
Lot of folks really love this. There are many characters I enjoyed but the way the main character, a teenage girl named Hitomi, frequently wussed and played the silly scared girl grated on my nerves (though not as badly as Minmei from RoboTech!). There are some GORGEOUS moments in this anime, though, so, I would recommend at least one viewing. And to be honest, I saw this in English... I wonder if I'd be happier with it in Japanese w/English subtitles? Worth a try once I get through all the other anime I have available to watch that I haven't seen yet... ;)
Witch Hunter Robin
LOVE this. It has a dark (but light?), mysterious feel, the music is awesome and the costumes... Just a neat take on 'witches' and a special organization designed to hunt them down along with more complex undercurrents and a few surprise twists and turns.
The Twelve Kingdoms
This took a while to get into but, once I had all the names and places and references down, very enjoyable. Seeing the primary character grow and come into her own was somewhat cathartic.
Card Captor Sakura
I was determined to hate this bubble gum, girly pink confection. My will is clearly weak. I LOVE Sakura. The brightness and love and optimism Sakura brings to every problem and relationship helped get me through some darker days when I needed positive things to focus on.
Chobits
CLAMP is awesome. This is funny, intriguing, somewhat mysterious as time goes on and touching too. Chi is a complete sweetheart and Hideki... well, Hideki is Hideki! :P Chi!
Trigun
Another "looks lighthearted but has some hidden meat/mystery to it" series. I'm typically not into the exaggerated, ridiculous overreaction style anime (of which there are parts in this series) but I found myself laughing anyway. The combination of futuristic-old west feel added kicks to this series.
Full Metal Alchemist
Wow. This is a bit heavier than the others and to think the premise is that too younger kids are going through all this is a bit heady. Some episodes are fairly lightweight but most have some lesson and/or moral overtone. And they're not afraid to kill primary characters...
Azumanga Daioh
Just fun, funny, lighthearted anime. The anime package info has great insights into explaining lots of elements of Japanese culture that are shown in each episode. We quote 'Osaka' in conversations many months later for her silly way of looking at things. And OMG the kitty outcome is SOOO cute! The male teacher just gives me the creeps though... *shudders*
Love Hina
More silly, teenage-ish fun. The poor male protagonist who's looking for a childhood love just keeps taking missteps around a group of fun and feisty girls who really let him have it on a regular basis.
Ah My Goddess! (first run only)
Another lightweight but amusing (and very pretty!) anime romp. A college boy makes a wish and winds up with a Goddess living with him - and humor ensues!
Serial Experiments Lain
This is NOT the typical animation style I like and the storytelling method is a bit disconcerting as well, actually. This series would bear re-watching as I'm certain there's still elements I just didn't get the first time through. Apparently this series has been the subject of a number of papers/articles discussing the philosophy of reality, the internet and more.
Magic Knight Rayearth (first series)
This one was a bit dated (CLAMP work) but fun anyway. Pu pu pu puuuuuuuu! (That's Mokona ;) )