Hi Cor-Caroli,
Glad to hear of your interest in anime style! As others have mentioned already anime style is a style - much like how Disney style is a style. Most of the people that get hired at places such as Disney, Pixar, Ghibli and whatnot first learn figure drawing and to draw realistically because they use that as a basis from which they "stylize" their art. I grew up in Japan and have drawn anime style all my life but I really noticed a boost in my "anime drawing skills" by learning the basics at art school.
As for books, I really like the series of books released by a Japanese animator called Toshi. They aren't translated, but I think it's still a pretty solid book series to learn the style of modern anime style from. It's sort of pricey, but you can probably find it at a local Kinokuniya bookstore if there's any of those in your area.
As for art materials - that's sort of up to the artist to decide what they like working with. I personally have been using a drafting pencil (this one you have to buy lead and a sharpner seperately). I find these way easier to draw with than a mechanical pencil and is basically the lead part of the pencil so it allows you to do blending and whatnot while giving you the ability to sharpen the thing into the pointiest point which allows you to draw some very fine lines with. It's super sweet. I also like sketchbooks made by Robert Bateman since they're nice and thick and inks don't bleed. If you want to try inking in manga style a manga artist favorite is the G-pen.
I would also suggest checking out Meetup.com to see if there's a local drawing group around you that has people that draw in anime style. Feel free to shoot me a message if you have any specific questions. ;)
Somone on the IRC recommended this http://www.amazon.com/Staedtler-Mars-Technical-Mechanical-Pencil/dp/B000YQEFGU/ref=pd_sim_sbs_229_3?ie=UTF8&refRID=085SH2ZWA6DGN1R739PH
Also look at a "lead holder" as opposed to a "pencil"
Like this:
It is very sharp and no wood.