If you can duplicate the database (ideally at a point in time when no-one is actively using JIRA) on to the new database, that's usually the easiest, fastest and cleanest way to do it. Obviously, you'll want to start with the same version of JIRA on the new server, then go through upgrading that and testing it all. Backup/restore is more for small systems and changes of database.
You'll need to copy the attachments, no matter what you do, but you don't strictly need them for technical testing or upgrades. A lot of people doing what you're doing will copy them initially, and then use something like rysnc to bring them up to date after testing.
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