Surnames are a rich and varied topic. In Latin countries, children generally bear their father’s surname first then their mother’s (with grandparents’ surnames sometimes thrown in). Tibetans have no surnames. A Tibetan may have one, two, or three names, which are often unisex but may be completely different from other family members. Some married couples give the mother’s maiden name as the child’s middle name. I have a niece who’s first and middle names are her maternal and paternal grandparents’ maiden names, respectively.
Between consenting adults, it’s up to the parents whose surname(s) appears on the birth certificate and in which order. It is perfectly reasonable to request that both names appear together, with or without a hyphen. The conventional thinking is that when both parents’ surnames are used, the man’s name usually goes first, then the woman’s. But there is no need to follow convention. If it is important to you that your name appear first, then you definitely need to make sure you are both in agreement on that point. If the order doesn’t matter to you, then you can leave it up to the other parent, come to a mutual agreement, or flip a coin. My business partner and I used a coin flip to decide the order of names for our business.