The first three apps handle essential account services and APIs, while the last app is the store. You'll install four applications: Google Account Manager, Google Services Framework, Google Play Services, and the Google Play Store. If you've ever installed Google software on an Amazon Fire Tablet, these steps should feel familiar. The next step involves downloading the proper APK files for the Play Store. You can browse F-Droid's library and download the app store from its official website. Still, it might have something specific you're looking for, like a FOSS fork of Telegram. Unlike the Amazon Appstore, you won't find social apps like Twitter or Instagram. It's composed of open source games and applications, so its selection is relatively minimal. One more alternative app store is F-Droid.
If you'd rather avoid those headaches and can live with the shortcomings Amazon's platform brings, download the Appstore. Still, this is an easy way around the frustration that trying to get Google Play Services up and running can bring.
