How do I prevent end users from being notified that their ticket was received?
In order to do that, you will need to modify the default trigger Notify requester of received request by adding the condition Other: Current user is agent.
Here is an example:
The admin can also go ahead and change the email body of the notification.
Please note that when you deactivate the trigger fully, agents will not be able to send out a message from Zendesk Support at all.