The 'right' thing to do for your MySQL databases is either to use the CLI 'mysqldump', or phpMyAdmin, and export all 'your' databases (don't worry about the MySQL control databases - unless you have added users to MySQL). Its easiest to do a move with the files stored in .sql format, then you can see what is in them or edit if needed. If you use phpMyAdmin, make sure the "Add IF NOT EXISTS" box is checked, makes it easier to restore.http://dev.mysql.com/doc/refman/5.1/en/mysqldump.html
Also - Save your config,inc.php for phpMyAdmin.
When you are on your new machine, you will be able to import those SQL files, using mysql CLI or phpMyAdmin.
As for the rest - save all the files ad folders YOU created under 'my sites' or htdocs (where ever your server root is).