I wouldn't normally teach someone the basics but I'm feeling generous.
An FTP app it used to copy files from one location to another using one or more file transfer protocols. We don't need to go into how they work but just that they do. We recommend the free and superb
Filezilla for the task. It is IMHO the best app of its type. It rarely fails unlike many other apps we have tried. Please try it.
Once you have this installed you can copy your website to your desktop from your hosting space or vice versa. If you have a load of images it may take a while so you have plenty of time to read up on these things.
The second app you will need is phpmyadmin (or similar provided by your hosting package). Coppermine not only stores stuff in files but also in a database. This database is not accessable on a normal setup via ftp so via phpmyadmin you can take a copy of your database into a file on your desktop. This file can then be restored to a new database. If your host doesn't provide phpmyadmin or similar tool then you will either have to ask them for a backup or once again the coppermine team come to your rescue with a copy that can be installed as a plugin to coppermine -
http://forum.coppermine-gallery.net/index.php/topic,60908.0.html . There is also a version for 1.4.x if you look.
Once you have both those things it is simple to move to a new host. It is also important for you to use those tools regularly and take backups regularly. Things can go wrong with hosting. They get hacked, files get corrupt and we've even seen some go bust. Especially with free hosts your support is limited so if the worst happened with your data where would you be without that backup? So it is important for you to learn these tools.
One other benefit of taking a backup in such a way is that using another free tool such as
Wamp you can run a copy of your website on your desktop. That way you can try upgrading or modifications first and get it clear in your mind before doing it on your live site.
The above is just for reference. If you have any questions on those tools use Google as I do not have the time to teach those basics. I really cannot stress how important it is to take backups so please follow my advice.