Reading through this, I have to comment on the difference in attitudes toward anime in and out of Japan.

Western fans tend to like anime. Japanese fans tend to like specific shows.

I have used this example countless times, but it`s kind of like, well, liking magazines. Anime is a media form - not a genre. Saying "I like anime" is along the lines of saying "I like magazines!"... Or "I like television programs!" It is incredibly broad, and sounds kind of stupid when you think about it.

But for western fans, it seems that anime is a genre. "If it`s anime, I know I`ll like it" sort of thing.

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Back to the suggestions though.

Instead of trying to pick up and watch a show with the intention of finding some anime you like.... Why not do what most Japanese (who watch anime "casually") do?
Look at the listing for the upcoming/current season of anime, see if you can spot something that doesn`t look unappealing, check out what kind of story it has on wiki... And if it sounds like it might be interesting, try watching the first episode or two. If you think it`s good, keep watching. If you don`t, stop.

For the summer lineup, we`ll be continuing to watch 夏目友人帳(going into the third season now?) and checking out 2 or 3 new ones that might be decent.
It`s just like any other type of programming. If it looks decent, watch the first episode or two and see if it`s something you would want to keep watching.

That's why I asked for a general overview of a variety of kinds that anyone would enjoy. I know there are a lot of niches and genres but I don't even know where to begin. I don't have the "I enjoy any anime" mentality because I've never watched any.
By gaining suggestions, I can begin to shape what I would like to see more of, and then ultimately narrow it down to what style I like, getting away from your hypothesis.

What I've noticed is we use it in two ways "Which anime" and "What kind of anime".
The 2nd kind is more like how Japanese people use it, it parallels "What kind of movie do you like?"

The 1st way is more like "Which CD do you like?"

I agree with your prognosis of the western view of anime.
Perhaps for a long time there wasn't much selection available so it was either all or nothing. Just a guess.

I've always thought that anime was no different than your average TV show. It just has happens to be animated. It has everything a TV show has anyways...

Fullmetal Alchemist is a great anime with such a wonderful story!
I also recommend some of the shounen anime such as Naruto, One Piece, Bleach, Katekyo Hitman Reborn.
I'm also currently watching Fairy Tail which is an epic, humorous show about mages ^^
I hope you like them and I hope I helped! :D

We'd be curious to what you watched. Please report after you sampled some good anime.
:D

I think it becomes overwhelming to answer which anime is good or worthwhile for another person to watch knowing that my tastes range a lot.

If you think of movie genres in general what are you drawn to? Action, comedy, serious drama, period pieces, realtionships, fantasy, sci-fi, crime? At least that would be a good starting point.

I would recommend Mushi-shi, Azumanga-Daioh, Honey & Clover, Saishū Heiki Kanojo (aka Saikano), Samurai Champloo, Berserk, Full Metal Alchemist, Death Note, Ghost in the Shell (movies and tv series), xxxHolic, & Gurren Lagann off the top of my head. I am a big fan of Neon Genesis Evangelion but I don't think any of my recommendations are similar to it at all.