Download button on music app

Good afternoon Mariuuus,

Welcome to the Apple Support Communities!

I understand you are trying to download albums and playlists from Apple Music to your Mac. The resources below will help you with this.

Add and download music from Apple Music or buy music from the iTunes Store

"On your computer

1. Open the Music app or iTunes. 
2. Find music that you want to add to your library.
3. To add a song to your library, click the add button . To add an entire album or playlist, click +Add.
4. To download music so you can listen to it offline, add it to your library. Then click the download button . Or find a playlist in the sidebar and click the download button .
5. If you want to access your music library across all of your devices, turn on Sync Library."

Create and share Apple Music playlists

"By default, the songs in the playlist stay separate from your main library. If you want to add each song in the playlist to your library, you can change your settings."

On your Mac

1. Open the Apple Music app.
2. In the menu bar, choose Music > Preferences.
3. Click the Advanced tab.
4. Deselect "Add songs to Library when adding to playlists."
5. Click OK.

Source: Add a song to a playlist in Apple Music without adding it to your library

Add music to iPhone and listen offline - iPhone User Guide

"Add music from Apple Music to your library

Do any of the following:

• Touch and hold a song, album, playlist, or video, then tap Add to Library.
• When viewing the contents of an album or playlist, tap Add to add the album or playlist, or tap  to add individual songs.
• On the Now Playing screen, tap , then tap Add to Library.

To delete music from the library, touch and hold the song, album, playlist, or music video, then tap Delete from Library.

Music you add to iPhone is also added to other devices where you’re signed in to the iTunes Store and App Store using the same Apple ID and have iCloud Music Library turned on (go to Settings , then tap Music)."

Take care.