What is the appropriate percentage of money to withhold when withdrawing money out of your 401K early for both federal and state taxes in the state of Georgia?

This is not state specific. A plan administrator is required to withhold 20% of the distribution which is sent to the IRS to be applied towards your tax liability.

I don't know of any state that requires withholding for local income taxes.

When you file your return, you will have to pay a flat 10% penalty, plus the gross amount of the distribution will be added to your taxable income for the year to be used for calculating federal, state and local taxes.