I just installed zenphoto so still learning my way around. Here is my install:
Zenphoto version 1.4.6 [bbddab7675] (Official build)
WEB path: www.themoehrings.com/gallery
Current gallery theme: Basic
PHP version: 5.3.27
Graphics support: PHP GD library bundled (2.1.0 compatible)
supporting: gif, jpg, jpeg, png, bmp
PHP memory limit: 128M (Note: Your server might allocate less!)
MySQLi version: 5.5.15

The setup files were not able to create the .htaccess or robots file and I had to manually change permissions and add folders. So it seems like there may be a permissions problem.
I got through setup finally, but cannot save any theme options. The page says they were saved, but it reverts to default.

Also, if I try to change the theme, the thumbnails no longer work.

Any ideas?

First thing to do is to look in your Debug log.
Admin -> Logs (tab)

Also, if your server does not support mod_rewrite
(that's why a .htacces file is not created) make sure the mod_rewrite checkbox is not checked.
Admin -> Options -> General -> URL options -> mod rewrite

[EDIT] Some info about permissions:
http://www.zenphoto.org/news/permissions-for-zenphoto-files-and-folders

Thanks for the reply. I have mod rewrite turned off. I tried a fresh install at www.themoehrings.com/zp/ It still doesn't save theme options. Are those stored in the db or in a special file that I should check the permissions on?

Thanks again.

Looks like you are on a Windows server.
Have had similar problems on a Windows server too.
Turned out the host had way too many security restrictions configured.
Try contacting your host.

Everything is working except for the theme options. I'm able to change all other settings. . . when I attempt to change them, it says changes were applied, but it doesn't change. So I don't get any errors.

Did you look into the server error log? That might have errors the Zenphoto own log doesn't.