K9Mail with Fastmail, AddressToGPS + OSMAnd, I back up my own photos with a home server and some rsync (Syncopoli). There is PhotoBackup which is kind of neat though.
We use Google Calendar for stuff at work, Google still has an old psuedo-deprecated-not-talked-about CalDav endpoint I hit with DavDroid that gives me all of that. These problems were solved with protocols long before gardens. I use offlineCalendar(Yes you need an app for that in Android) to create a calendar for myself locally, though I have a radiCAL server for things I need sync'd. Could have used Fastmail I guess.
Funny, I use exactly the same apps on my Cyanogenmod phone - K9 + Fastmail, OSMAnd. There arnt that many good alternatives to Google, apparently.
The only thing I'm still missing is public transit navigation.
I do have the Play Store installed though - through OpenGApps pico - I only have the Play Store, none of the other Google stuff - with a dummy gmail account.
My city has a transit app they maintain, but I'm afraid that'll stop if Google gets too ubiquitous in this area. It's really too bad that information that should be released as clean openData sets is so difficult to acquire and parse easily. Really should be some regulation around what, how and how often public information should be published and maintained.
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.
I am using "Here WeGo" https://here.com/ (originally developed by Nokia), quite a good alternative to Google Maps. It does not need any Google services installed. Downloaded it directly from the Google Play store with Raccoon.
Hmm, that Raccoon thing seems ok but using something like that would be a bit of a sideways move for me. I'd rather move toward 100% libre software and not rely on proprietary software I'll need to hack around to maintain privacy.
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.
pse: thanks all AP a lot for suggestions :)