You do not need to install Joomla first. Here is what you should do to transfer your site to another server:

1. Copy all files from your current server to the new server in binary mode. It is very important to do it in binary mode.
2. Make sure that the major version of PHP is the same on the two servers. Go Daddy has an option to set PHP on version 5.2 or 5.3. Check what version your 
    current server is on and set that version on Go Daddy.
3. Create a backup of your database on your existing server. There is only one database for Joomla and you can find out about it by checking the configuration.php
    file of Joomla by looking for $db = ....
4. Create a new database on your Go Daddy account and restore your database backup file there.
5. Open your configurations.php file and edit these variables to match the new server:

$user
$db
$pass
$log_path
$tmp_path

After this you should be able to see your site as it is on the new server.