BEFORE you change your phone number, you will need to obtain backup codes to regain access to your account. Backup codes are located in the google settings under the 2 factor settings. Print them out and keep them in a safe place.
Visit https://myaccount.google.com/security, click the expansion arrow to the right of “2-Step Verification”, and then log in with your [email protected] account.
Inside the 2-Step Verification section you will see an option for backup codes, as well as an option to update your phone number. Click the expansion arrow on the right to view and print your backup codes.

After regaining access to your mail by using a backup code, you must now update 2-factor authentication with the new phone number.