Anyways, if you can't go wrong with the classics:
Laputa
Nausicaa
Spirited Away
Lupin III: Castle Cagliostro
These all are available in sub or English dub format.
Well, if you're looking for the most accessible anime, you probably want one of the big three or similar shows. If don't know what I'm talking about, the big three are Naruto, One Piece, and Bleach. Similarly long and action-y shows would probably be Dragon Ball Z, YuYu Hakusho, Fist of the North Star, or Fairy Tail. Now, all of these shows are pretty long (I think Naruto's pushing about 550ish episodes by this point), so these shows are a bit of an investment.
Of course, there are shorter heavily action-based shows, like Black Lagoon, Jormungand, Deadman Wonderland, Elfen Lied, Higurashi no Naku Koro ni, Hellsing Ultimate, and so on. But these and others like it can be /very/ violent (Elfen Lied, Hellsing Ultimate, Black Lagoon, Higurashi), or simply mediocre (Deadman Wonderland).
Side note: Full Metal Alchemist, both the 2003 and 2009 versions. is an excellent story with a relatively limited amount of episodes (around 60 for both), that is highly recommended by just about everyone.
There are other good anime (for beginners) as well, depending on your taste in genre. Baccano! is an action-packed show, with a great (but often confusing) plot, and excellent characters. Or there's Fullmetal Panic, a show that combines mecha action with high-school comedy (Note: Season Two focuses heavily on the action and plot, while the 'joke' Fumoffu season is heavily comedy oriented). Or The Melancholy of Haruhi Suzumiya, which launched a slew of high-school comedy shows; and, while I despise those, I do very much enjoy the original. Or (last suggestion, I promise) Steins;Gate, an excellent show about time travel and its consequences.
The shows I've named are but a drop in the bucket of what you can find (trust me, there are about thirty more shows I'd love to talk about in depth on here). Ask around, see what others recommend, and keep an open mind. Don't force yourself to like a show because you think liking anime necessitates liking everything within. There's a lot of crap. If you don't like it at all, you can move on.
Good luck! I hope you find that you really enjoy some of the shows in this medium.
Edit: If you want to watch the shows with english dubbing, the legal way about it would be to sign up for a paid account on Funimation's website (Which has most, but not all of the shows I mentioned on it available for streaming). Since you're new, it would probably be more cost-effective to just search '[Anime Title] English dubbed', and watch an illegal streamed version. Not a habit I would recommend to keep if you do really get into anime, if for no other reason than the fact that you should support the industry so it can continue to provide us with wonderful anime.
hmm.. something like a group of super heroes or people with powers taking out evil people or a group of bad guys but drawn by Japanese people? If so some of the easier animes to "get into" which im assuming you mean easier to follow and more enticing than others at least for me would be. Fairy tail, Naruto, Soul Eater, DBZ (Dragon Ball Z) in order from newest to oldest. It also highly depends on what you're into, if you like magic and stuff definitely fairy tail, if you're into import cars and stuff try initial D as some examples
I would suggest if you're into action shows to watch something like Naruto, One Piece, Bleach, Dragon Ball Z, or Soul Eater. I'm not really into these shows (The only one I've watched is Soul Eater.) but if you're into the action stuff that is in Teen Titans and Justice League, you might like those shows.
You can't just "get" into something. You have to already have some sort of interest in it.
you can google "superhero anime". and maybe check out this:
http://en.wikipedia.org/wiki/Category:Su...
I really hope I helped
i use to be into cartoons when I was younger but want to get into anime no subtitle stuff I'm not reading but I want to get into something similar like teen titans justice league but in anime form easiest way to get into it from stuff like that