"Way" is very general. "That's the best way to do it." "Method" is usually used to describe something that has specific steps, like a scientific experiment. "This is the method you must follow to get results." To be "methodical" means to carefully repeat the same action over and over again. "Manner" is usually used to describe a behavior, but it's also formal. "She walked in a strange manner." You will hear "way" most often in conversation.

Cathy gives a great explanation!