What are the general materials needed to be an anime artist?

Learn Copic Markers. Become familiar with the different thickness types they have, and play with them!

There are programs for Manga filters and effects to make life easy for you. Or if you're on a shoestring budget, you can use Microsoft Paint program to create effects, by painting shapes with certain colors to manipulate types of shading when you convert the piece in monochrome/B&W.

And lastly, learn to draw background. As much as many artists are concerned with human anatomy, background tell the story. After that, it's easy street.