If you want to migrate your website from J1.5 to J2.5/3.x or if you need to move your AcyMailing data from one website to another, you should:
1/ Install the latest version of AcyMailing on your old website (that will make the migration easier)
2/ Copy the folder media/com_acymailing from your old website to your new website
3/ Move all the jos_acymailing_* tables from one website to another (you should first export the database tables via phpMyAdmin and then import them on your new website)
Be careful, you may have to change your table prefix from jos_ (or another one) to the one you use on your new website.
So once exported, you should open the file and replace the text "jos_acymailing" by your new prefix (for example mnpo_acymailing) and then import the file on your new database.
4/ Install AcyMailing on your new website
DELETE FROM #__acymailing_config WHERE namekey LIKE "acl_%"
Don't forget to replace #__ by your own table prefix.
Migrate with jUpgrade
There is an AcyMailing plugin included in jUpgrade since jUpgrade 2.5.1 so the database tables and media folder migration will be handled by jUpgrade.
If you use jUpgrade to migrate your website, the only thing you will have to do is to re-install AcyMailing on your new website, all your data will be already migrated by jUpgrade.
Migrate with SP Upgrade
SP Upgrade also takes care of the AcyMailing data migration.
Just make sure you install the latest version of AcyMailing on your old website before the migration, then trigger the migration via SP Upgrade and once done, reinstall AcyMailing on your new website.