Hello bgalvan2,
You wonder about using Apple Music without the need to stream the songs using your data or Wi-Fi.
The music is stored in the cloud but you do have the option of downloading the songs you like for offline listening. This will take data for the initial download but you can then listen to them locally without using more data.
See
Add music from the Apple Music catalog to your library on your iPhone, iPad, iPod touch, Mac, or PC - Apple Support
You can save songs, albums, or playlists to listen to when you're not connected to the Internet.* You must save songs, albums, or playlists on each device that you want to use for offline listening.
*You can save music offline on up to 10 devices. These songs are accessible only during your Apple Music membership.
- In the Music app, find the item that you want to save.
- Tap the More Options icon
to the right of the song, album, or playlist name.
- Tap Make Available Offline.
Songs and albums appear under My Music > Library. Playlists appear under My Music > Playlists.
Best regards.