In cPanel, under the Files heading, there may be a Backups icon. Click it, and there will be a button labeled Download or Generate a Full Website Backup. Click that, and it will let you download a cPanel backup of your entire site, including mail and databases. You can then give that backup file to your new host to restore on the new server.
The server administrator does have the option to disable the Backups feature in your cPanel, so it may not be available to you. In that case, you would need to log into your site via FTP, download your site data, and upload the site data on the new host's server. Your new host should be able to help you with this.
