I added a new ticket and sent the customer an email as a public reply, but they replied with an internal note. Does this mean that customers can see my internal notes?
No, customers can't see your private notes (see Adding comments to tickets).
However, if the update is made by a user who isn't on the CC list or isn't the requester of the ticket, the update is displayed as an internal note.
For more information about the expected behavior, see Understanding when email replies become public or private comments.