If you are wholly unfamiliar with anime, take the ones commonly touted as insane or "weird" with caution until you are. They may be praised the most, but that's because those anime are mostly built for anime fans and their idiosyncrasies can easily turn away those looking to get into it. It's my belief that a lot of anime made in recent years is like this, with a worrying majority of them emphasizing crass sexual humor even in non-comedy shows.

You should at least familiarize yourself with the classics (Dragon Ball, Sailor Moon, Pokemon, Gundam, etc.) as well as those titles that have been recommended in this thread the most often. You're not going to like all of them, but they embody many of the basic tropes that most anime associate themselves with. After that, I would recommend any anime that those who generally hate anime like. This includes Cowboy Bebop, Ghost in the Shell, Fullmetal Alchemist, Samurai Champloo, etc. These anime appeal to the serious crowd while still being fairly humorous in their own right. Once you get a feel of how mature and plot-driven anime can be, the funnier stuff can even it out. The titles already noted in this thread can point you to the notable comedy shows. If you want artsy stuff, the Studio Ghibli movies and the series Mushi-shi are great examples that illustrate how nuanced Japanese animation can be. Dubs and subs are purely subjective and aren't worth discussing in this forum, but my rule is to give all dubs a chance. Not all dubs are any good, but not all Japanese voice tracks are better than the English ones. My stance on the sub part of the argument is that most people who prefer subs believe that dubs sacrifice fidelity to the original meaning by necessity if they don't just hate the voices. Some people hate the English voices for the former reason, which isn't fair to them. Those people don't realize that subtitles do the same thing, so if the original experience is so important to them, they should turn off the subs and learn the original language.

My personal recommendations include:

• Azumanga Daioh: A comedy anime that many haters enjoy. It set the standard for the "group of high schoolers talk about stuff and do nothing" style of anime that persists to this day.
  
• Magical Girl Lyrical Nanoha: It's boring in the first four episodes of the first series, but everything that follows is totally worth it. It subverts many concepts of the magical girl genre and makes magic fights really cool to watch.
  
• Mushi-shi: It's very low-key and subtle in all the right ways. A lot of anime coming out these days makes it harder and harder for me to believe that subtlety isn't dead in anime. If you liked the atmosphere and attention to background detail in Princess Mononoke, then this is that in episodic form. Pay attention to the music; no two episodes have the same score.
  
• Strawberry Marshmallow: Another comedy show. Four elementary school girls and their older sister figure actually get stuff done. Be warned that a few people think it's for pedos, but I don't see the connection.
  
• Digi Charat (the original series, which is hard to find): My all-time favorite show. Not for everyone, since it requires some knowledge of the culture.
  
• Cowboy Bebop: It was on Adult Swim and popularized the mature bands of anime in the West. The dub is pretty good, too.
  
• Trigun: Same reason as above, but to a lesser extent.
  
• FLCL: Same reason as above, but for a different reason. It's six episodes long, so you can appreciate it for its brevity, but the incredibly fast pace can eat you alive if you don't pay attention. Only watch this after getting some experience. I hold the English dub as a gold standard to which all other dubs are judged; even today it's just that good.
  
• Cromartie High School: High school tough guys experience weird things. For a show about delinquents, there's very little fighting.
  
• Rumbling Hearts: Very, very dramatic show, known to reduce grown men to tears. One of the most celebrated drama series that highlights the fragility of the human psyche in the face of a great loss.
  
• Love Hina: The first popular romantic comedy of the previous decade. The sexual humor here is tame compared to today's anime, but it was considered outrageous back in its day.
  
• Slayers: A classic. Even the newer series, Revolution and Evolution-R, manage to keep the old-school charm that made the originals fun to watch. It's one of the few anime whose filler is worth watching.
  
• Read Or Die: On the surface, it sounds like a show about an international conspiracy regarding codes hidden in classical literature, when in reality it can be described as magic fights with living pieces of paper as weapons. The TV series is easier to swallow than the three-episode OVA, but it's to be watched afterward. I recommend the dub to this one.
  
• Galaxy Angel: A poor adaptation of a video game series turns out to be pretty funny. The anime dropped its space opera roots for a gag series of hyper-incompetent military excavators and it worked out so well that many people are surprised it's even based on a game.
  
• Shin-chan: A rare example of a gag dub done right. It was originally a children's show, but as is it wouldn't have much chance, so it was transformed into a mature show for Adult Swim rife with American pop culture references and more poop jokes than were in the original.
  
• Sgt. Frog: Another dub in the same spirit as Shin-chan, but much closer to the original script. It's high on built-in references, but the dub glosses over most of them.

Finally, absolutely stay away from The Melancholy of Haruhi Suzumiya at all costs.

In case you're wondering, Excel Saga is for professionals. Episode 26 is the final exam. I would only recommend it to the brave.

Kable wrote:

I also realized that most of the titles we mention are all kinda complex for new commer in the anime world and we don't know if he can asimilate it the same way we did. In my case I watched anime since I was 12 and now I'm 21 so if you trow something like exel saga or in a lower level chobits I'll understand it right away and know how to apresiate it. If one wants to enter in the world of anime, one must start with dragon ball... completly understandable battles of unlimited power and epic awesomness...

I don't know if there are any ideal anime for newcomers, but they certainly don't include most recent shows. So many of them are wrought with sexual humor and most of the good anime don't deserve to be represented that way. There's only so many directions you can go after comically large breasts and pathetic dudes who upend skirts accidentally or otherwise. To make things worse, I have heard more than one person recommend Neon Genesis Evangelion for one's first anime, which I think is one of the worst shows to recommend for how dense it is. It's this kind of myopia which seems to be exclusive to anime fanbases that makes it so difficult to start talking about it today.

If you plan on only watching one bit of anime, make it this:

Or if your impression of it wasn't destroyed by countless live sketches inspired by it.

Drew Linky wrote:

Has someone said Negima yet? Yeah, Negima.

Be warned that the ending to the first series was so awful, they remade it six times, none of which have been licensed in America.

All of them after the first series follow some part of the manga. The first series ran out of time and budget and scraped together a huge cop-out ending, but followed it somewhat up to that point.