Do I need to download iTunes app to my iPhone 6 or is that what the music icon is?

You do not need to, nor can you download iTunes on an iPhone. iTunes is only for computers.

On an iOS device such as an iPhone there are several apps / icons that offer iTunes functionality and features.

To buy music, movies, Tv shows, you use the iTunes Store App:

To view, manage, and play music you've bought from the iTunes store, or synced from a coputer through the iTunes application there you can use the Music App:

Changes in the new iTunes - Apple Support

music can be found in your music app if you are managing it from another device or if you have purchased it through apple.

No need to download anything. Just tap the iTunes app in the Dock.