F-Droid (https://www.f-droid.org) is a free, open-source software repository alternative to Google Play. Much of the software Moonbeam.town recommends, such as K9 Mail and Element, can (and should!) be downloaded, installed, and managed directly from the F-Droid app. It is, obviously, Android-only; I'm honestly not sure if iOS has an equivalent, but given Apple's devotion to its walled garden, I rather doubt it.
That's it! You're now ready to search for and install software packages from F-Droid. Good ones to start with would be Element (for Matrix chat and K9 Mail (for Email).
Alright, so one thing about F-Droid is that it will never automatically update your software for you. You can adjust the app's settings to notify you about available updates, and even download those updates in the background, but you will always have to manually update them one-by-one. This is a feature, not a bug – by forcing you to approve each update, F-Droid reduces the chances of one of the apps you use being installed with some major change without your consent.
Well, you don't. You can install most if not all of these apps using the standard Google Play interface. But then Google would see you doing it, would know you had done it, and would know just a little bit more about you. Since reclaiming some element of our collective privacy is one of the points of Moonbeam.town, using F-Droid and non-Google software is one of the small steps we can take to service this ambition. With F-Droid, there is nothing tying you as an individual to the apps you download and install, and I think that's kinda nice.