What is the Islamic ruling on a wife traveling with her family without her husband's knowledge, and what is the correct action to take towards the wife and her father who assisted her in this? And is informing the wife's father (about her travel) considered permission for her?

It is obligatory for the wife to obey her husband in what is permissible. It is his right that she not leave the house except with his permission. His permission is what suffices, not the permission of his father or her father. The husband's obedience takes precedence over the obedience of parents, as Ibn Taymiyyah said: "When a woman marries, her husband has more authority over her than her parents, and her obedience to her husband is more obligatory upon her." A wife traveling with her father without the husband's permission is not permissible. It is advisable to remind the wife of the obligation of obedience. If she does not respond, the means of reconciliation mentioned in Surat An-Nisa should be employed with her: {As for those [wives] from whom you fear defiance, [first] advise them; [then if they persist], forsake them in bed; and [finally], strike them. But if they obey you [thereafter], seek no means against them. Indeed, Allah is ever Exalted and Grand.} This should be done while considering the nature of women, as stated in the Hadith: "And treat women kindly, for woman was created from a rib, and the most crooked part of the rib is its upper part. If you try to straighten it, you will break it, and if you leave it, it will remain crooked. So treat women kindly."