Side-load apps
You will have to enable the Unknown sources option in the settings for side-loading apps. Whenever I have to configure a new device, I install all the apps (mostly games) that I need from Google Play before disabling it and all its associated apps. I then use Total Commander (a file manager app) to copy APK installer files of the apps to an SD card backup store. If ever I need to install those apps again, I can get them from the backup.
There are many sites that provide free downloads of APK files. Installing apps from these sources is fraught with danger. AndroidPolice.com maintains a site at www. apkmirror.com where the latest APK builds are uploaded for testing. XDA Forums is a site famous for its Android developer community. Many APK files that are not available on app stores can be downloaded from here. Always read the reviews before installing any APK.
Racoon is a Google Play desktop client. It is a Java software that you can run on Linux, Windows or Mac computers. It does not break Google’s ToC. You have to log in to a valid Google account to download the APK files, which can then be manually installed on your phone. There is no need to have Google Play on the device.
In my opinion, F-Droid is the best alternative to all the rest.