Well, the first step to anime is in realizing that it is no ordinary cartoon, and most fans refrain from calling these works of art mere "cartoons". In these instances, they feel this way because they have had the time to absorb it in and appriciate it.
The reason for this is because japanese animation, by it's very nature, is multi-themed, multi-classed, multi-genred and multifaceted. There is anime for intellectuals and children alike.
The anime industry has just as many genres as the movie industry does. You name it, anime has it.
I think though that in order to begin to like anime you have to understand it. And for an american, such as myself, it requires that you look at it from more than one point of view. Remember, anime was not made for you or for me, it was made for the japanese people and so is often tailored to them and their culture. Albeit, anime is more universal today than it was since it's humble beginnings. So remember at first that it may be difficult to adjust to what you might normally be used to. It can be much more deep and uninteresting at first, but if you stick with a series it will usually grow on you in some way.
Anime resolves heavily around plots, stories, and characters much like a movie would. These characters are often well thought out in advance and take a life of their own. One of the magical things about anime characters is they are so easily identifiable with pretty much anyone of any mentality. So characters are, more often than not, memorable and easily become fan favorites.
Anime is also about the music. Alot of anime has some wonderful BGM or soundtrack additions. The genred music in anime is just as diverse as anything else can be. Theres rock, pop, metal, techno, jazz, blues, and even classical, and more. Music is very important in anime and is a great addition to the telling of a story or the mood of a situation, or the sort of theme or demographic the series is playing to.
And lets not forget the artwork itself. It is of a level and quality that easily surpasses even some of the best western styles. Just looking at some of the artwork is very inspiring and realistic and plays just as much to your soul and mood as it does the story.
These are just some of the many things that make anime so great, better than regular cartoons.
Now, to actually get into anime, I'd suggest making friends, online and off, that you know are into anime. Anime fans are usually refered to as "otakus", meaning "anime fan". There are thousands of webforums dedicated to anime in general and specific anime series as well.
To get the best experience possible from it however, and the most knowledge, its probably best to actually go to a library or bookstore and look for reference material. I know for a fact that there are books that basically tell you about the history of anime itself and how that got started and where its going.
Most bookstores will also sell "manga". If you don't know what this is, its the japanese equivilent to comic books. There are plenty of online sites you can purchase manga and anime from as well. My all time favorite is Robert's Anime Corner Store, I've been doing business with them for over 4 years, and its just excellent service, excellent selection and excellent prices. They also sell misc. things as well and apparel.
If you're looking into watching an anime series I really couldn't recommend a good one unless I knew what you would like. I don't want to stereotype or assume, but it looks to me like you have a liking for gothic, vampiric, etc. sort of things. In that case, theres plenty anime and manga to wet your appetite with.
Vampire Hunter D is an interesting short film. I personally think it's overrated, but its a good start looking into this genre.
Witch Hunter Robin is a very good one that was only fully released recently, and it is available in a complete dvd boxset.
Theres plenty of others, and you could search them by genre at your own leisure.
I'm not sure what else to say at this point. Normally I'd have all the answers and it would have been presented in a better way, but this thread caught be off guard.
Feel free to ask questions. I'm sure someone will have the answers for you. ^___^
