If your new server have cpanel, you can backup and restore on new server. Also Cpanel offer easy site transfer feature when logged in as "root", you can ask your web host or server admin to see if they can use WHM Transfer feature to do the site move. This is easy as entering your domain name, existing user name and password, click transfer button, in most case, it will work.

Other way is download files, upload to new server. Backup MySQL, restore on new server. Now set all file permissions, edit include/config.php, update the database and path (if changed)