Anime is met with mixed attitudes by the western audience. Some aren't able to get over the stigma that animation is for kids. Some find the situations too ridiculous (often without realizing that one of the advantages of animation is that ridiculous situations are made possible without needing expensive special effects). Some have simply never been exposed to anime at all. Most people, however, just watched an episode of Dragon Ball Z, thought "Wow, this is stupid," and vowed never to watch anime ever again. Dragon Ball Z is a fine anime, but much in the same way that Sex and The City is a fine TV show. It has a very specific audience that it caters to very well.
What I'm trying to say is that little effort has been put forth by publishers to bring anime to the wider audience that it deserves. There is an anime out there for people of all ages, genders, ideologies, and personal tastes. Here I have compiled a list of Starter Anime. The list isn't in order of best to worst in terms of entertainment or artistic quality, it is ordered best to worst in order of "How easy will an anime hater become engrossed in this?" I haven't included any sort of links, but there are a variety of ways to find these shows either subtitled or dubbed into English. If you are willing to tolerate it, the general consensus is that the original Japanese performances are superior. Exceptional dubs are noted in the list. The list itself is inspired by the first one I talk about, Death Note, which I believe is something that everyone should see, everyone can enjoy, and everyone can really learn something from.
Death Note - Animated purely by coincidence. This is a suspenseful, thought-provoking, dramatic two-season series that would already be nominated for awards if it had been released on America's HBO. The Death Note is what the Shinigami (imagine a society of Grim Reapers) use to end the lives of mortals. Whoever's name is written in the note shall die. This is normal when the power is wielded by the Shinigami, as it keeps the wheel of life & death turning, but what would happen if a Death Note fell into the hands of a human? To add interest to intrigue, what if this person was extraordinarily ambitious, intelligent, convicted in his beliefs, and absolutely unafraid of killing people in the name of justice and the greater good? Watch this show with a friend, preferably someone with a very different personality. You'll be arguing over the moral ambiguity for weeks.
Akira - This one takes a step deeper into traditional Japanimated Weirdness, but it's considered a classic by all who have seen it. Anime writers love giving incredible power to unlikely individuals, Akira is the prime example of this practice. Tetsuo is your typical teenage delinquent. He sits at the bottom rung of his gang, he's jealous of everyone around him, he desperately wants to prove that he's a tough guy worthy of attention, and he's willing to go to great lengths to get that attention. He is tired of Kaneda, his long-time friend and also the leader of the gang, bailing him out of trouble and talking down to him. Then, suddenly, his powerful psychic abilities begin to bloom. College-aged Peter Parker was a smart man able to understand that with great power comes great responsibility, but teenaged Tetsuo is an entirely different story... (Good English Dub!)
Gantz - Gantz is the game of death. Death for most people means just that: Death. But for some, it means waking up in strange room with other recently deceased people, being given strange weapons and instructions, and being forced to fight for another chance at life. This is a story of confusion and gradual revelations, so I can't say much more than that. I will say, however, that unlike many televised programs, Gantz is not cheap with its characters. Watching your loved ones die is just par for the course in the cruel world of Gantz, perhaps that is even part of the purpose of its existence.
Cowboy Bebop - I love Cowboy Bebop. I had to put it further down the list for a few simple reasons... It includes rough & tumble bounty hunters traversing a futuristic galaxy with their space ship, the Bebop. There is Spike, the protagonist with a dark past, a master of Jeet Kun Do and former member of an assassin's syndicate. Jet, the gruff pilot, former police officer (with a somewhat less dark past), and apparently skilled chef. Faye, the scantily clad token female with (of course!) a dark past that she doesn't remember due to amnesia. Ed, the happy-go-lucky hyper little girl who just happens to be the best computer hacker in the universe. And finally, Ein, the super intelligent dog. This show has nothing to do with cowboys. Despite how it sounds, Cowboy Bebop is dead serious and very heavy. The fact that it manages to suspend disbelief in such a setting with such over-the-top characters makes it all the more enticing. If you can get absorbed into it, you will be well on your way to being a full-fledged anime fan. (Good English Dub!)
Neon Genesis Evangelion - Now we're into the territory of what makes anime a love-it-or-hate-it style. This show follows the lives of three youngsters who have had the fate of the world thrust upon their shoulders. There are giant robots. The story is convoluted and confusing, possibly requiring multiple viewings to fully understand. The characters have various neuroses, making them anything but predictable, and certainly not logical. The story is loosely based on those of the Christian bible, though interpreted through the eyes of non-Christian Japanese writers. There are multiple endings, all of which are valid according to the writer and director. Amongst lovers of anime, this series receives almost universal praise for it's deep plot, extraordinarily dynamic character development, and incredible insight into the human mind. Amongst everyone else, this series receives blank stares, looks of bewilderment, and outright furious confusion. If you can appreciate it, then you are an anime person. Go nuts.
If you get through the list without vomiting, then you should probably start looking for anime that suits your personal tastes. Do you like sex & violence? Check out Ninja Scroll. Do you like inexplicable cuteness? Chi's Sweet Home and Azumanga Daioh! are for you. Are you looking for juvenile action guilty pleasures? Naruto and Dragon Ball Z were made with this in mind. What about chick flicks? Go for Chobits (or anything published by Clamp). Now I'm certainly not trying to make the argument that all anime is good, or even that anime is for everyone. Just like every other published media, anime is hit-or-miss, generally with more misses than hits. However, there are many more excellent anime out there that I haven't mentioned here, and they are designed for a very diverse set of tastes. The problem is that they tend to all be classified as simply "anime," so one must dig a little deeper to find the "hits," and even more to find the ones that match your tastes. But if you manage to acquire the taste, it's definitely worth looking into!