What is the ruling of the Sharia regarding staying at the husband's family's home first due to a family custom, while the wife and her family insist on staying at the wife's family's home first?

Among the beautiful aspects of marital relations is for each spouse to treat the other's family with kindness, to help their spouse honor their parents and maintain ties with their kin, and to overlook their mistakes. Therefore, you should strive to reconcile your family with your wife's family.

Regarding your wife's visit, try to convince her family or your family to give up the condition that they be visited first. If that is not possible, there is no harm if your wife starts by visiting your family without her family's knowledge, or vice versa, even through indirect hints and dissimulation (tawriyah).

However, if that is not possible, your wife is not obligated to start by visiting your family, and you are not required to force her to do so.