well its good courtesy to hold the door for all who are following directly behind, and more so to go a bit out of the way for a lady. I was directly refering to a lady being with you. You open the door and hold for her. However more clearity on the matter is good.
If a lady opens the door before you can reach it and then holds it open for you. Upon reaching the door you should take the door and reply with a gracious thank you.

I have no faith in human perfectability. I think that human exertion will have no appreciable effect upon humanity. Man is now only more active - not more happy - nor more wise, than he was 6000 years ago. Edgar Allan Poe