Since the gallery (and the albums inside) will contain family photos, I would like the whole site to be accesed only by registered users.

Zenphoto is correctly installed and working on the index page. The webpage will solely have the gallery and its albums.

Regards

You can set Admin > Options > Gallery > [i]Gallery type[/i] to [i]private[/i]

You might also want to protect the /albums and /cache folder via htaccess as Zenphoto does not control direct access to the image itself.
http://www.zenphoto.org/news/htaccess-against-hotlinking/

If you really want to protect your site from outside access, the most secure way is to do this server side by using htaccess to password protect it:
http://www.htaccesstools.com/articles/password-protection/