What is the ruling on my obedience to my mother in taking her to visit her friends, given that my father does not approve of her leaving the house and she does so without his permission?

A mere separation of the spouses does not constitute a divorce, and it is not permissible for the wife to leave her husband's home without his permission, except out of necessity. Therefore, the mother's going out to visit her friends without the husband's permission is an act of disobedience and insubordination (nushooz), and it is not permissible for the son to assist her in this, for "there is no obedience to a creature in disobedience to Allah." It is advised to strive to reconcile between the parents through supplication and by seeking assistance from someone influential with them.