Fdroid for what I use daily. Firefox Aurora direct from FF which auto updates. HumbleBundle for games because I like funding DRMFree + outside GPlay distribution methods.

I use raccoon to download from GPlay if I need to as a peer stated and know when to update with apktrack from fdroid, but really that's only for Slack since we use it at work. Slack uses Google Push Notifications like most apps, so it doesn't notify me about updates, but I check it periodically enough. I could go the microG route, but I prefer not having to interface with google servers at all if possible.

Edit: Maybe I can either convince my work to use something like Riot.im when it gets mature enough, or use a slack bridge for it and rid myself of this problem.