1)方法:The method for obtaining something or achieving some intents.
为了追到那个女孩儿,他想了很多方法。
2)办法:The way for handling the things or solving the problems.
处理空气污染的办法。
3)方式: The manner, form or mode that people speaking or doing work. Mostly use 方式 to handle abstract things or or humanity.
他很快适应了中国的生活方式。