The way he talks makes me really annoyed.
Koichi's style of conversation is very cultured, eh?
Wow! The way you talk sounds just like a native Japanese person.
The way of speaking is your way of speaking (or it could be your speaking style). This is how someone talks. Some people have an interesting speaking style, after all.
The readings come from 話す and 方. By now you should have seen these words separately and should be able to read this word too.