Do I get free music for having Apple Music?

As long as your subscription is active , think of it as borrowing. You can listen to whatever you like but it’s not yours to keep once canceled.

use the music app to access Apple Music. The iTunes App is for purchases and if you buy a song you will be billed separately.

Apple Music Terms 

Manage your Apple Music subscription - Apple Support 

If you cancel Apple Music, you'll lose access to the Apple Music catalog, any music that you added to your Library, and the features of Apple Music when your current subscription period ends.

If you cancel during your trial period, you’ll lose access to the catalog and features of Apple Music immediately

You can listen for no additional charge as long as you have a subscription: Apple Music - Apple