If I gave a family member $15,000 as a gift, do I, the giver, have to pay taxes on that money or is it not taxed on either person?

The recipient of a gift never pays income tax. The giver of the gift is required to file a gift tax return form 709 if they give more than $15,000 per person per year. However, tax is not owed until their lifetime total of gifts and a state is more than $11 million. Form 709 is used to track your gift against your lifetime total.  If the gift is exactly $15,000 per person per year (or less) then form 709 is not required.

The receiver of the gift does not report the gift received on a tax return, regardless of the amount of gift received.

The giver of a gift does not report the gift given if the amount given to a single individual in a tax year is $15,000 or less.

The gift giver only has to report the gift given if the amount given is greater than $15,000 to a single individual.  This is reported on IRS Form 709, not part of a federal tax return.  There are no taxes owed on the gift if the aggregate of all gifts ever given is less than $11.58 million.