Use of the Play Store due to policy enforcements etc, for publishing there, is highly advisable and/or Aurora (Open Source Play Store front end https://gitlab.com/AuroraOSS/AuroraStore/-/releases) if you don't wish to use sandboxed play services in a particular profile.
The issue with this is that many apps will look for FCM for push and unfortunately many more in the messaging category will not have their own fall back solution/implementation.
The next best step is to go direct to the website of a developer ie Signal or direct from their Github repository ie Newpipe.
Both of those examples have a self updating mechanism to notify you of new releases.
For apps that don't have this feature take the URL of the releases page and append .atom to it and add it to your preferred RSS feed app with notifications turned on for updates.
This does have it's downsides creating a dependence on F-Droid sources below for apps such as Element Android who only publish their gplay version on GitHub as well as for apps published on gitlab without an atom feed to track.
The next option is to use Neostore to access F-Droid repositories but with an up to date high SDK targeting F-Droid store front. This allows for unattended updates also I believe and has a modern UI. It is a bit buggy but is constant and current development.
You can also if you choose download apps direct from the F-Droid website with the links they provide.
F-Droid however does come with some caveats explained here by @Wonderfall https://wonderfall.dev/fdroid-issues/
I also use a Hash Checker app to confirm I have the same app for when the developers provide a SHA256 hash etc to compare it to in their README on github.
Obviously moving forward GOS will be looking to add some self built third party apps that meet the projects standards to the Apps app built into the OS while there is also Accrescent being worked on in parallel as a broader app source looking to be the next evolution to the open source app store from @lberrymage found here: https://accrescent.app/
Hope that helps and look forward to others views and opinions.