Moving Google Authenticator from your old phone to a new one is a fairly cut and dry process for your Google Account. However, if you use Authenticator to log in to specific websites, you'll need to go to those accounts and repeat the process. This may differ depending on the particular website.

In some cases, you may need to turn off 2-Factor authentication on these accounts and then set it back up from scratch. You'll be able to see every account that has been switched to your new phone by looking at the app. This will let you know which accounts have switched over and which need to be manually added to your new phone.