Like Spotify and some other streaming music services, an important feature of Apple Music is the ability to temporarily save songs for when an Internet connection is unreliable — or simply to avoid data caps imposed by cellular carriers.
Before beginning, the most important thing is to turn on iCloud Music Library via the Music section of the iOS Settings app. This is required to add online songs and playlists to the My Music section of the iOS 8.4 Music app.
Be warned, though, that trying to turn on iCloud Music Library can potentially delete locally-created playlists and their associated files — from both iOS devices and a desktop iTunes library. Proceed with extreme caution and make sure a backup is in place.
At that point, however, saving music for offline listening is extremely simple. When browsing songs, albums, and playlists, triple dots (ellipses) will appear next to many items. Tapping the dots brings up a menu with a "Make Available Offline" option.
Once that's selected, the item or items in question should be saved on-device and searchable as if they were local content.
Files saved this way can be kept for as long as an Apple Music subscription is active. Once a subscription lapses, the cached files will disappear.