*shakes head at 99% of anime suggestions in this thread*
Okay, I'm going to suggest some things that are actually quality entertainment, and you can walk away from feeling really good about *except Lain...you will walk away feeling like you just dropped a LOT of acid*
Kino's Journey
In a word: beautiful. The perfect image of humanity, faults and all. A bit of a culture study, you might say. However, it has great characters, interesting situations, and a beautiful art style.
Haibane Renmei
Another brilliant series. Heavily relying on religious themes without actually attaching itself to any particular faith. Great characters, and it's based off of a Yoshitoshi ABe manga. He did the design for...
Serial Experiments Lain
This series may cause your head to explode. Heavy in philosophical statements about reality. Extremely disturbing at times, but engaging nonetheless. I suggest this to people who don't even watch anime because of the challenging content. Don't let yourself miss this series.
Beck: Mongolian Chop Squad
Music anime. No joke. It's about a band. Pretty vulgar, so if language bothers ya, you may want to look elsewhere. It has a great story until the final episodes, where the director of the show decided to rush the ending instead of waiting for the manga to finish.
Crest/Banner of the Stars
Very dramatic sci-fi series. Nothing to do with mechs either. Unique for what it is, it's centered more around a slow moving story line, so it's definetly not for everyone. The english dub is terrible too. I enjoyed it simply because it does have an engaging storyline because of it's realism whilst being in the realm of science fiction.
Cowboy Bebop
Yes, this is the complete opposite of everything you asked for. But, keep in mind, I wouldn't suggest it if it wasn't a masterpiece. If nothing else watch the following episodes:
Ballad for Fallen Angels
Jupiter Jazz 1&2
Hard Luck Woman
The Real Folk Blues 1&2
If you don't at least have some emotional response to the end of Hard Luck Woman, you officially have no soul. I'm serious.
Last Exile
Another masterpiece. Twenty six episodes of steam punk glory. Great characters, and a very deep storyline. I lie to you not, this was a trip I would love to take again...if only I had the remaining dvds of the series. Range Murata's art style was perfect for the settings in the show. If you are unfamiliar with his work, check out the artbook/manga 'Robot' (mind you, that is an 17+, and has some stuff that I don't read because...well...it's kinda sick. Some of it is just bizzare in a sexual sense, while not actually being sexual).
Voices of a Distant Star
This is a short film from Makoto Shinkai, and it is one of the most emotionally devastation films you will ever see. The story is of two lovers, who are torn by war. One stays on earth, as the other moves further and further away from him through space. They try to stay in touch with text messaging on their phones, but it takes more and more time with each message. Brilliant in style and execution. No violence. No language. No sex even. A bonus on the dvd is an even shorter film called 'She and Her Cat', which is a poetic piece which I recommend just because it's poignant view of human beings from those who see a more open side of a person: the pet.
I know I sound like a jerk in the opening of this post, but really, I can't stand 95% of the anime that gets released. It's all so similar that it's sickening. I look for series that are unique, interesting, intelligent, and have characters that I can enjoy seeing the stories of. I despise the cookie cutter crap that most anime 'fans' enjoy.