OK, so I have Dreamhost who oftens ZenPhoto "easy" but apparently that's just a shared environment. Unfortunately my client set up her site on the easy site and now I have a real full install and I want to move all her uploaded albums and photos (100s of them) to the new site. What advice do you guys have to do this painlessly?

Use the Zenphoto backup facility to make a backup of the database. It will be created in the "backups" folder.

FTP the Zenphoto files to the new site. (You should use the same release you are running on Dreamhost.)

FTP the albums to the new site albums folder.

FTP the backup file to the backups folder

Run setup. Select the restore option rather than the create admin option when setup is finished.

Just to note there is an entry on the troubleshooting on that as well:
http://www.zenphoto.org/news/troubleshooting-zenphoto#40