well, if you just backup all databases from mysql that should work (including the mysql database which actually handles all the permissions on access to mysql).
Then you also need to backup the the Mail folders as well as the folders for the website. Make sure you keep the correct permissions.
Well, I don't have ISPConfig (yet) so this is what I think what needs to be done but be carefull:
Once you have the backup of all ISPConfig related stuff then you must set it up back on the new server. Once ISPConfig is working there, then you can put the backup files back on the new server. Hence extracting the mysql dbs, the mail folders and webfolders. Important thing - in my opinion - is that you keep all the permissions right.
However, don't turn your old server off before you didn't check that the new one works right