1. Backup manually the old course and download the resulting Zip
2. Open the file moodle.xml in the root directory of the Zip backup and see if there are any absolute links (eg http://mydomain.com/file.php/32/). Now open just that one file in a text editor, and do a FindAndReplace to replace all occurrences of "http://mydomain.com/file.php/32/" with "$@FILEPHP@$". Save the moodle.xml file and overwrite the old moodle.xml with this new one.
3. On the new Moodle, create a new course and then visit that course and note the number in all the URLs for that course - eg if the course URLs are like this: http://mydomain.com/files/index.php?id=999 it means that all the data for this course is stored in the moodledata folder called 999
4. Upload the backup file to the new moodle using FTP (or any other method) into this specific folder /moodledata/999
5. Visit your new course and do a 'Restore' using the Administration menu within the course
Write any other tips/tricks to doing a course backup to help others