I would like to make a site with content that anonymous users can read, and private content that only group members can read.
How do I achieve this? I have installed Group, but any group content I add can be viewed by anonymous users.
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Some basics about he Group module to be aware of:
gnode, a submodule of the Group module.
It seems that you would want to grant access to what, in Group, is called "Members" (similar to what in your question is called 'Group members'). But you woudn't allow access to "Anonymous" (site visitors that are not logged it). However, you also need to think about which access you want to give to "Outsiders" (= logged in users who are not a member of a group).
With the above in place, it's just a matter of assigning a node to a Group: based on how you configured the roles and permissions, the Group module will allowed/deny access for any user.
Refer to "How to create collections of content that can be accessed by users with various access levels?" for more info about all this.