Yes, you can start your Microsoft 365 Personal subscription and then upgrade it to a Family subscription. When you upgrade from a Personal subscription to a Family subscription, the remaining time on your Personal subscription will be converted into a prorated credit towards the Family subscription.
The exact cost of the upgrade will depend on the current pricing of the Family subscription at the time you decide to upgrade. Typically, the cost of the Family subscription is higher than that of the Personal subscription, so you would need to pay the difference after applying the prorated credit from your Personal subscription.
For specific pricing details, it is best to check the Microsoft 365 subscription options directly on the Microsoft website or through your Microsoft account management portal.