anime is just short for Japanese Animation, if you were to describe an anime style you would have to describe the person who designed your said anime like Miyazaki or Satoshi Kon... you'll want to start taking classes in traditional animation and figure drawing at some point if you're serious about anime.
Anime is a English word meaning a Japanese cartoon. Not all Japanese cartoons are drawn in the same style, however as there has been in the past an exchange of culture before, there still is one so sometimes the East and West exchange ideas of methodology and style of drawing.
Alternatively, Anime is a Japanese word describing any animated show. Sailor Moon is anime, the Amazing Spiderman is anime, Tintin is anime...
Also untrue. A lot of famous artist love anime. Notably, Murakami. Who's entire career is pretty much built of satires of it. The problem is not with liking anime. The problem is being an idiot and not being open to other ideas and not practicing your fundamentals. Some people think you can learn a style of art. No you can't. That comes with habitual practices. Learn to do the grape vine before you salsa.
If you want to learn how to draw anything involving the human form, the best thing to learn and understand would be realistic human anatomy Regardless of what style you're looking to improve in, understanding the human body and how it works will make it more reasonable to develop your own stylistic take, and better exaggerate the form without it looking like a technical error.
As for what classes you need to take, anything that teaches you cartooning and realism. Unless you are in japan, which offers specific courses on anime and manga, you are better off learning to draw correctly.
Unfortunately, this is something I'm willing to bet you don't want to hear.
Actually no, as a musician, classically trained in piano and violin, I can kind of relate. Being classically trained helps me with all my techno music just because of the fact that I know what I'm doing when it comes to playing on those instruments and what key signatures I want. So I totally want to take a traditional route of learning art and getting it right the first time.
Anime is a Japanese abbreviation of animation, so Japanese cartoons. So anime = cartooning. Cartooning are a deliberate simplification of reality for effect. Often but not always applied to areas where ideas need to be communicated in an economic way repeatedly. When you hear art teachers or others heave and frustrate over anime its more about the attitudes of some students that think they don't need to study the fundamentals because they draw "anime". So if this doesn't apply to you then just rock on and do your best!