First make sure to install and run the latest version of Yummy FTP, available from the website, then activate it using the registered Name and Serial Number that was emailed to you after purchase.
Quit Yummy, then from your old installation...
Mac HD -> Users -> yourname -> Library -> Preferences -> Yummy FTP -> Bookmarks
Mac HD -> Users -> yourname -> Library -> Preferences -> Yummy FTP -> Item Information Database
Mac HD -> Users -> yourname -> Library -> Preferences -> com.yummysoftware.yummy-ftp.plist
... to the same locations in your new installation.
If you're moving these files across a network, make sure to zip them first using Create Archive from the Finder's File menu, then expand them at their new location. This will ensure all the data is transferred intact.
Keep an eye out for v1.8.4 coming out in the very near future...