To move to a new computer, simply download the latest version from our web site and install BeerSmith on your new computer.
You can retrieve your key from your old computer by going to the "Enter Registration Key" command on the Help menu.
To move over your recipes or other data, go to your old copy and export anything you want (such as all of your recipes) to a BSM file using the Export command from the File menu. You can do this from My Recipes view or any of the Hops, Grains, other views depending on what you wish to export.
On the new install just use the Import and Merge into the appropriate view to import the BSM files you saved from your old version.
When you install the new copy, I highly recommend installing to a subdirectory under your Documents directory for the main login you use. This avoids a whole host of permissions problems that can crop up.
Cheers,
Brad