Mysqldump is good, here's the manual page and a tutorial page
http://dev.mysql.com/doc/mysql/en/mysqldump.html
http://www.devarticles.com/c/a/MySQL/Backing-Up-Your-MySQL-Databases-With-MySQLDump/
But if you've got phpmyadmin on the sites it's got a dump feature you can use to grab a zip or .gz or plain dump of the database.
Restoring it is as simple as pasting or uploading the dump (if it's small enough) or using
mysql databasename < databasedump.sql
On the command line to restore data from the file.
