Matt,
For best results, do a complete dump of your Listmail database with PhpMyAdmin and save it to your hard drive.
Set up a new database, username, and password on the new host, and use PhpMyAdmin to re-import your data.
Upload ListMail to the new host, entering the new MySQL access information into config.php. Login to ListMail. Set up email aliases, cron tasks, etc.
Hope that helps!