How does my family member log onto Apple Music or iTunes

If you have an Apple Music Family subscription and have set up Family Sharing, then each of the group members get access to the Apple Music subscription content, each through their own Apple ID account, with their own preferences and library and playlists.

Note that Apple Music Individual subscriptions are not shareable in a Family Sharing setup. You’ll have to change the subscription type for that. About Apple Music plans - Apple

About Family Sharing - Apple

Learn about what you can do with Family Sharing - Apple Support

Set up Family Sharing - Apple Support

If you meant something different, then please elaborate.

They don't have to log in at all anywhere.

Once you have invited them to the family group and have shared the Apple Music subscription with them, it should automatically be available to them in their own Apple Music app on their own devices.

Click here ➜ Add a family member to your shared subscriptions - Apple Support