iPod touch

It sounds like the songs have not been downloaded to the device yet. Using the new Music app - Apple Support

Listen offline - Copied from Above Link

You can download songs—including songs that you bought from the iTunes Store, added from Apple Music, or that were matched or uploaded with Apple Music or iTunes Match—to your device to listen to when you're not connected to the Internet:

1. Tap the More Options icon next to the item.
2. Tap the download icon to the right of the song's artwork at the top of the More Options menu, or tap Download from the bottom of the menu.
3. The song downloads to your device.

To see music that you downloaded to listen to offline, go to My Music. Under the Recently Added section, tap the category selector (Artists, Albums, etc.). Only Downloaded Music is at the bottom of the category list. When this setting is on, My Music shows only songs that you downloaded to your device. Turn this setting off to see all of your music, including music stored in iCloud Music Library.When you view an item that you've downloaded to your device, you'll see an offline icon near it, but its exact location varies. For example:

• If you download an entire album or playlist, the offline icon appears when you're looking at the album or playlist in My Music. You'll also see an offline icon when you view a compilation, but not when you view specific genres, artists, or composers.
• If you download a song, the offline icon appears to the right of each song title.
• If you download songs from an album, but not the entire album, the offline icon appears to the right of just the songs that you downloaded.