Lessons
At the moment there are hundreds of lessons that gradually and teach to portray certain elements in anime style. This includes not only the eyes, hair, clothes, but also the environment, landscapes, and composition. First, go to as many such lessons. Pay special attention to the image of people, as it is the Foundation of any figure in the anime genre.
Please note that each author portrays the characters differently. Of course, there are similarities, but the differences still more. Therefore, to completely copy someone's style is not worth it. Try to keep only the General outlines of the kind of expressive eyes and bright colors.
See video tutorials. Please note the order in which authors create figures and how they hold the instrument. Try not to neglect minor details because they make the drawings really good, and the characters expressive.
Practice
Once you learn normally represent separate elements or parts of a drawing, start creating your own characters. Consider all the elements: from hair to shoes. Be sure to carefully choose colors. They should be bright and at the same time realistic.
On anime forums, often held competitions among beginners and experienced artists. There you can exhibit their work, receive adequate critique and even to any prize. Anime festivals are also often held such events, but the competition is much stronger.
Creating some good characters, try to draw own comic. First, it is sufficient to use a 3-4 frame. Come up with some simple plot and try to make sense of the characters. You can use special programs like MangaStudio, which significantly facilitate the process of drawing comics.
If you want to achieve great heights in drawing anime, put their works in Japanese and English resources. There are really experienced artists will give you specific recommendations. Moreover, many publishers are viewing such forums, looking for talented artists. Who knows, maybe they will pay attention to you.