In my previous question I asked about how Google Play Store is updated. And this question is sort of an extension to the previous one. What updates what? I know that Google Play Store updates other Google Services (like Google Games, Kiosk, Music itp.). Does Google Play Store update Google Play Services or does Google Play Services updates Google Play Store or do both of them updates themselves?
Google Play Store is the only one capable of downloading and installing apps. All apps including Store itself are updated via Store. As I've already answered, the update is automatically downloaded and installed as soon as it's detected by Store.
All of those other Google play apps (services, music, books, etc.) are just like any other app on your phone, when Google releases a update for them, Google play Store downloads the update and installs it when you are not using the app.