I am trying to prevent members of a group from being able to see who else is in that group.
So, one idea is to have multiple groups, where each group would grant 'read' access to the site. The 'Set Up Groups For This Site' permissions menu lets you set 1 group for read access. And you can't add groups to groups.
So, is there any way to prevent visitors from seeing all other visitors?
Yes, you can hide the links to the people and groups, but if they know to goto _layouts/people.aspx?membershipGroupId=x, they can still view.
And yes, you can limit permissions on the people.aspx page through windows, but this does not help when the site uses FBA and admins need access to the page.