Well, PSE Organizer has catalog files that it creates if you in fact organize through Organizer. When I upgraded from 5 to 6, I had no trouble converting the catalog for my new version of Elements. It's not retrograde, though, you can't convert from 6 to 5, or 7 to 6. Once you convert the catalog file, it's converted.
If you just copy the files, the tags won't necessarily go. The catalog file is what you will have to put on your new computer. You can literally copy and paste the file, you just have to find where it lives. I moved mine from the default location to my EHD so that I can access it and back it up whenever I need to.
HTH!
Sarah
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Sarah 
Equipment: Sony A200, 18-70mm, 75-300mm
Software: PSE 6.0
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