Subway - means subway trolley, or subway train. Yes, we have fast food store (that sells sandwiches or subs basically) called "Subway". But if you don't mention food, you say subway, people usually understand you are talking about public transportation.
Underground - it is clearly mean something underground: pipes, facilities, subway train running underground. By the way, we have subway trains running underground and on the street.
Metro - I guess it means Metropolitan. In other words, big city.
This is based on my best knowledge. I live in New England area, east coast of the United States.