A couple of months ago I made a couple of (guilty) in-app purchases for a game, under the impression that even though my family has Family Sharing, that my details would still be used for the purchases. I was later informed that they had in fact been charged to my mum's card (the one who's details are in Family Sharing) instead of mine. Is there a way to set it up so that my details are used instead of my mum's, so I don't have to use her money and can instead use my own?
Family Sharing uses a single credit card for all users on linked account. You cannot change this on an account by account basis.
You can purchase and redeem an iTunes Store gift card and add that to your account. iTunes and the App Store should automatically pay for the item with your store credit before using the family credit card.