I have been watching anime since I was a little kid. I was lucky enough to not have to differentiate between Western and Eastern animation when I grew up. One of the first cartoons I remember watching was La Abejita Maya (Maja the Honey Bee – Mitsubachi Maja no Boken) when I was probably 4 or 5. The series is very well written and always had a good message at the end of the day.
Back then I had no idea I was watching Japanese animation because to me everything was considered cartoons. Some kids experienced the same thing here with a couple of series like Astro Boy and Speed Racer, but overall Japanese animation was not really popular in the US until about 5 years ago when the market started getting saturated with some very bad anime.
Anime is a hate it or love it kind of thing for a lot of people. I don’t really see the point on totally disliking it but there are a lot of people out there that just despise it. I totally respect their point of view. Anime has been linked to the negative geek label now, but I would only put fanboys on that category. The internet makes it even worse, trying to find information about anime can leave you with a headache because most websites that follow it are poorly written and have the most obnoxious colors available in the RGB world.
There is a lot of good adult anime, that rivals in story depth with some of the best hollywood has to offer. I think Spirited Away showed anime in a different light that some people might not have expected. If you have never really experienced anime I think this is a good starting point. I would steer you away from the recommendation most anime fans give you as a start which is Akira. While the movie is excellent it is quite complicated and not in a good way.
Anime is far reaching when it comes to subjects. You can find anything from animal series that are for kids to adult geared hentai that tends to be a little too bizarre for my personal taste. Just remember that there is a lot in between. My favorite genre of anime is the post apocaliptic future Tokyo type of theme. A lot of times this series involves humanoid robots that develop a soul, in what I believe a heavy Asimov influence ala Ghost in the Shell. Or robots trying to fit into our society like the movie Armitage III of the series by the same name.
Also from growing up I hold a series dear to me, Captain Tsubasa or Supercampeones as it is knows in Spanish. This soccer fueled series falls prey of the repetitive animation and impossible physics that sometimes turns people off, but I just love it. It is a great series and there are a lot of people that still follow it today. I have most of the original series dubbed in Spanish, and have always wished that they release it here in the US. I was hoping that with the popularity of soccer in the US, they might bring the latest iteration (Road to 2002) but I was not that lucky.
The last series that I want to mention is Rurouni Kenshin, which I own the whole series on DVD. This anime has a lot of history about Japan and how the country changed when Samurai became a thing of the past. Very similar in some aspects to the recent Tom Cruise movie The Last Samurai. While it might not be actual history, it does open a window of at least interest to a time when Japan was changing from being complete feudal to become the industrialized nation it is today.
Anime has a lot of flavors and while some might be quite bitter, there are also some sweet flavors that are worth trying. I think it can be a good experience as trying ethnic food, but of course with a little guidance. I say when looking for anime try to find something that might mix something you are already interested in. My bet is that one or more of your hobbies have an anime already linked to it. Just don’t let a bad experience keep you from trying something else in the menu.