Setting up non-Admin user account in Drupal / CiviCRM - Menu bar not visible
One option would be to create a Menu Block or just a Block with some links and set it in to a region with settings so it only shows for Authenticated

But also, am unclear if you really want a non-admin person getting to those 'modules'. 

If a non-admin person is making donations, registering for events they do not need access to CiviCRM itself.

Perhaps update your Q with more detail about what the non-admin person needs to access perhaps by providing the paths you want them to get to so you avoid saying 'member page' which could be construed in many ways. HTH