If you are on an iOS device, you don't need the "Music" app to buy songs; there is a dedicated storefront. Open the app (the purple icon with a star). Search for the song or album you want.

Buying music from the iTunes Store without actually using the iTunes app (or its successor, the Music app on macOS) is straightforward thanks to Apple's web-based and mobile alternatives.

You can browse and purchase music directly through a browser on any device. Navigate to the Apple Music Web Player. Sign in with your Apple ID.

The music will then appear in your library across your devices.