The Apple Music app will operate much like the iTunes app. You can buy music for super cheap and your old music should still be there. Just be sure to backup your music files soon just in case something happens. Apple Music carries a monthly subscription fee if you want to stream music (kind of like Spotify) but you can listen to the music you own without paying anything. Everything is essentially the same, but it's been broken up into three apps. One for music, one for movies, and one for podcasts.
It's possible to download CD quality music to your phone fairly easily, but when using the standard AirPods that come with the phone, I personally can't tell a difference in how they sound.