This article will show you how to download music from Google Play Music to your Windows or Mac computer. This only applies to purchased and pre-uploaded songs to your Google Play Music account.
Steps
Method 1 of 2: Google Play Music Site

Step 1. Go to this address
music.google.com.
Sign in to your Google account. If you're not automatically signed in, please enter your email address and password

Step 2. Go to the Music Library tab
This tab is located in the column to the left and looks like a stack of records with a musical note at the top.

Step 3. Click on the Tracks tab at the top of the page, below the search bar

Step 4. Click on ⋮
Hover your cursor over a song or album and click on the three vertical dots icon that appears in the upper right corner.

Step 5. Click on Download or Download the album.
If you don't have this option, you may not have download rights for this song. Click Buy to buy this song
Method 2 of 2: Using the Download Manager

Step 1. Go to the Download Manager address
Open your browser and go to https://play.google.com/music/listen?u=0#/manager to download Download Manager.

Step 2. Click on Install Download Manager
It's an orange button at the bottom of the page. After that, the download of the "Download Manager" will begin.

Step 3. Double click on the setup file to install the Boot Manager
On Windows, follow all directions to complete the installation. On a Mac, move the Download Manager icon to the Applications folder to complete the installation.
By default, the installed files can be found in the Downloads folder

Step 4. Launch "Boot Manager"
Double-click the orange earphone icon in the Recently Added section of the Start menu (on Windows) or in the Applications folder (on Mac).

Step 5. Log in to your Google account
Enter your Gmail address and password to sign in to the account you will be using to work with Google Music.

Step 6. Select Download songs from Google Play to my computer
Click the "Download songs from Google Play to my computer" radio button to select it, and then click "Continue".

Step 7. Select a folder to download the song
Choose a default music folder, or click the Select Folder radio button to choose a different folder.

Step 8. Select the music you want to download
Click on the Download My Library button to download everything you have in this service, including the songs you downloaded. Alternatively, you can click on Download My Free and Purchased Songs to download only songs that you have purchased or received for free from the Google Play Store.

Step 9. Click on Start Download
After that, the music will start downloading to your computer.