In Gravity Forms that is called a "Notification" - See the article Confirmation v/s Notification
Start with something simple for your first notification. Fill it out much like you would your confirmation.
Fill in all of the basic information that is needed.
If you want to add more info (like the information from the form) put it in the customer message that you want to send out and use the drop down to pull the form fields into the notification
On a different but related topic: If you are using a form to bill members please remember that an authorize.net feed must be set up for each form implemented, see Authorize.net Integration
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