Each application on F-Droid has to be approved. If an application on F-Droid has proprietary code, the code is removed by the F-Droid maintainers. If an application contains tracking code, it is reported on the app page. In my opinion, you can guarantee that 99.9999% of applications on F-Droid will be safe to install. On Google Play, you can't guarantee it. Even though Google Play is the default Android marketplace, some malicious developers have been able to put malware on there.
F-Droid also contains applications that Google have banned from their store, such as Ad blockers.
The installation of F-Droid is like any other app that is not in the Google Play store. In order to install it, you need to enable unknown sources. You also need to keep unknown sources enabled in order to install applications from F-Droid.
You will get a warning that enabling unknown sources is dangerous. It's not. Unless you go searching for random apk files on the internet that may contain malware, you should be fine. Just stick to trusted sources like Google Play, Amazon Appstore, and of course F-Droid. Please note that F-Droid is safer than Google Play or Amazon Appstore.
In order to enable unknown sources, open the Settings app, and tap "Security". You can enable unknown sources from there. You will get a warning. Just press ok.
Now go to https://f-droid.org/ on your phone. Tap "Download F-Droid". It will download an apk file. It should be named FDroid.apk. You should be able to install it from your notifications. If that doesn't work, open your downloads folder in your favourite file manager and install the apk from there.
Video clip on how to install F-Droid
Video clip on how to use and install apps via F-Droid
There are plenty of applications available for download. Of course, not as much as on Google Play, but way less junk, and much safer thanks to the open source nature of things.
I hope you have found this useful. Remember, there's more to Android than just Google Play!