Q: Does Apple Music eat up a bunch of data?

It eats up as much data as you use. Generally your average song, depending on length, will be between 3-8MB. You can estimate your usage based on that. Some data may also be used by the Music app to download album artwork/images/etc, but this is usually a one-time process. The cellular data tracking in Settings.app will give you a good idea of your Apple Music data usage.