Manner = to do something which can be compared to another person' way of doing something, "In the manner of a drunken man". Way = Strict guidelines or practices, steps to follow. "Do it my way or the highway!" (do as I say or walk away). Method = to perform a task following accepted guidelines, or to do something in steps 1,2,3 etc "The Rhythm method of contraception" haha I see your confusion, these small details are very difficult areas of learning to speak English fluently - extremely fluently.

They are synonyms. This is the way to go (road, route). This is the way to get that job done. Do it this way. We have a method we follow to fabricate this object. Our method is different from the way other companies do it. A method could be a prescribed procedure that is followed. The word "manner" is common, but we might not use this word as much. It means "way" English has many words that mean the same thing, and many are interchangeable. You would not be wrong to use any of these words.