First, you want to make sure that both of the servers use the same version of Moodle. I would recommend updating (if you haven't already) the home machine server to 1.2. If your hosting machine is also 1.2, this process is a breeze.
Go into your course (in the home machine), choose "course backup" from the admin menu on the left, backup the course (including users). Moodle will zip all of the information into a zip file. Go to "Files" in the course admin menu, click on the backup you just made and download it to your computer.
Go to your new installation, upload the zip file from your computer to the hosting machine (by using the Site Files link in the new installation). When the upload is complete, you'll notice that next to it (in the uploaded files menu), you'll see an action to "restore" the course. Click on the restore link and follow the directions to either restore the backup to a new course or an existing course on the new installation. Voila. You should have a complete copy.
This feature was added by super Eloy, one of the Moodle developers.
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