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.

You can use Raccon http://www.onyxbits.de/raccoon : "Download apps directly from Google Play. Raccoon is the only APK Downloader that also supports paid and large apps."

I've got a few crutches I'm not ready to give up so I'm on regular Android but I've been starting down the path of using F-Droid only apps so I can trasition more smoothly when I'm ready. The only app I've got left is Maps, OsmAnd is a little too tedious for me but I'll convince myself it's worth it eventually. I'm also concerned that I might not be able to get Project Fi working quite right.

F-Droid and apkmirror. You can use any app that doesn't run google services, including open source alternatives for apps like youtube and google maps. Can access many things from your web browser, as well.

I use F-droid for many apps, but I also still use Google Play.