Installing F-Droid
F-Droid is a free, open-source app store for Android — every app in its main repository is built from source and verified to be free/open-source software, and it's the actual index this site's own Android catalog syncs from.
1Download the F-Droid APK
Since F-Droid is itself an APK not distributed through Google Play, you install it the same way as any sideloaded app -- directly from its own official site.
2Allow installation from this source
Android will prompt to allow installs from whichever browser or file manager you used to open the downloaded APK -- approve it for this one install.
F-Droid's own APK is signed and its checksum is published on the official site if you want to verify the download before installing.
3Open F-Droid and update its index
On first launch, F-Droid downloads its app index -- browse, search, and install any listed app directly, with F-Droid itself handling future updates for anything installed through it.