Is it considered breaking an oath if the younger sister went with the mother to visit the grandmother, after the older sister swore to her sister not to go as a punishment for her, knowing that the mother was aware of the oath and the one who swore it immediately sought forgiveness after swearing?

If by your oath you intended that the girl should not leave on her own accord, then you do not break your oath if her mother takes her out. However, if you intended that she should not leave at all, and her mother took her out, then you have broken your oath and a kaffarah (expiation) for the oath is incumbent upon you. The kaffarah for an oath is: freeing a slave, or feeding ten poor people, or clothing them. If you cannot find any of these, then fasting for three days. Your seeking forgiveness after the oath has no effect, unlike saying: "Insha'Allah (if Allah wills)".