I think Apple Music and iTunes are different things, Apple Music is the streaming service and iTunes is the store. If you have an Apple device like an iPhone, then the iTunes Store and the Music app would already be downloaded. If you want to do it from a Windows PC then you probably need to download the iTunes app, which contains both the store and the library.
While I’m not sure exactly how it is in your region with your favorite bands, some labels/bands only open their music to Japan. In that case, you would need to make a Japanese iTunes account which requires a Japanese address to make, also you would need to buy Japanese iTunes gift cards from online retailers because the system doesn’t accept foreign credit cards. If your music is sold in your region then you can just go ahead with a normal iTunes account and use your credit card. If you download music from your library, it will be stored as a file (forgot the extension sorry) on your computer that you can transfer anywhere. If your phone is an iPhone, then you can sync it to your phone’s library. I don’t know if there’s a cap of how many devices you can sync to a computer, but you can only sync one account from one computer onto a phone so be careful. However you can buy and store the purchases from multiple accounts onto a single iTunes library on your computer and you can purchase items from an account on your phone that’s different to the one on your computer.
I just had my wisdom teeth pulled so I’m kinda out of it, if you need me to explain anything better, just ask.