If my family has family sharing, do I have to use the family credit card for every purchase, or can I buy some things on my own for my personal use and other things through the family sharing?
Once you have family sharing turned on all purchases that aren't covered by a store-credit get billed through the shared payment method which is one charge card entered by the "Family Organizer" Apple ID.
Quote from the Apple Support article:
After you set up your family, any time a family member initiates a new purchase it will be billed directly to your account unless that family member has gift or store credit.
As a workaround @bmike recommends purchasing a gift card and/or refilling your iTunes balance so that no purchase ever is made on anything but store credit.
Discounted gift cards can be found at Costco and you can also store your iTunes account in iOS wallet app and recharge it at Apple Retail Stores.
The entire family’s purchases come from one account. The benefit to this approach is centralized control over purchases.
Once you have set up Family Sharing all your applications can be installed by other members of the family. I believe only the parents can pay for the applications unless credit is on the account. Other members can only 'ask' to buy applications.
You can find out more information about Family sharing here: https://www.apple.com/uk/icloud/family-sharing/