Do I need Apple Music to save songs offline?

Hi AgentTBone,

Apple Music is essentially a music streaming service. If you want to listen to music from Apple Music while offline, yes, you need Apple Music.

The other ways to listen to music offline is to sync your own music from a computer with OS X or Windows to your iPhone (see Sync your iPhone, iPad, or iPod touch with iTunes on your computer using USB - Apple Support), to purchase new music on iTunes, or to download music you previously purchased on iTunes (see Download your past purchases - Apple Support).

I hope this helps!

Bakhtina

The Phone icon means its downloaded to the iPhone and can be listened to offline.

If the other songs are not downloaded they'll need to be downloaded to the iPhone by tapping on the cloud icon.

Are you using iTunes match?

What happens if you try using the iTunes Store app (not the Music app) to download your past purchases?

Quoting Download your past purchases - Apple Support :

On an iPhone or iPod touch

1. Open the iTunes Store app.
2. At the bottom of the screen, tap More > Purchased > Music.
3. Tap "Not on This iPhone" or "Not on This iPod."
4. Find the song that you want to download, then tap it.
5. Tap the Download icon to the right of the title. Your song downloads to your device.
   

On an iPad

1. Open the iTunes Store app.
2. Tap Purchased.
3. Tap "Not on This iPad."
4. Find the song that you want to download, then tap it.
5. Tap the Download icon to the right of the title. Your song downloads to your device.

But you purchased it from the iTunes Store using this very Apple ID? Are you certain it's not just hidden?

Hide and unhide iTunes Store, App Store, or iBooks Store purchases - Apple Support

Did you actually purchase this song, tap on the Buy button for a price?

Or did you simply download it from the Apple Music Service?

What exactly did you do?

Do you have the three dots icon next to the song name?

It should look like this :

Image taken from this article : Add music from the Apple Music catalog to your library on your iPhone, iPad, iPod touch, Mac, or PC - Apple Support