The host that you tranfer to could do this for you, sometimes they do it for free, but if you want to try it your self you can use phpAdmin to backup your database, you should use that and make a back up even if you use the SQL plug-in too, ether way save the file, then ftp the entire contents of your web directory to a folder on your local computer, this should include the backup file for the database.
The host that you transfer to has to have all the proper settings and configuration that you had with the prior host, then ftp the contents of that folder on your local computer to the web directory on the new host, on the new host go to the cPanel create a database the same name as the database from the old host for the store, locate the file you save from phpAdmin and restore the database, this is the hard part you have to create the database user that the store uses in it's config files and grant premissions.
Then ether restore a copy of zc_install folder and run a upgrade or edit the two files /includes/configure.php and /admin/includes/configue.php editing the lines that reffer to that path to the web directory that would have changed with the host.
I think I got it, if I did not some one will catch it.
pete