[ONE General] 5chdn:
I'm using a Google-free device with maximum privacy, so maybe I can not answer all your questions but I can give you an idea. First of all - disclaimer: I'm here because my girlfriend has an Oneplus One (OPO), but I do not have one. I use her old Nexus 5 (N5), but you will get the general idea. You already noticed there is a difference between CyanogenOS (COS) and CyanogenMod (CM). It also took me a while to figure out that difference. If you still have a stock Android in your OPO, it should be a COS 13.1 which is based on Android 6.0.1 and comes with alot of bloatware from Google and Microsoft.

1. First step is to find a suitable ROM for your needs. If you are used to COS and have not much experience in chosing custom ROMs, you should give CM a try. Here is the official wiki which includes Download links and installation instructions: https://wiki.cyanogenmod.org/w/Bacon_Info

2. The 2nd step after installing CM is the full device encryption, can be found in Settings > Security. If you do it on a clean phone without any apps and data it only takes a couple of minutes and chances of breaking stuff are low at this step.

3. Now I usually root it (with SuperSU) and install some magic which is called XPOSED framework. It's something which allows you to install modules on your phone on system level, not like an Appstore, but rather like a Tweakstore. There are a lot if chances you break stuff and most of the modules do not work with CM, however, one module to beat them all is the XPRIVACY module. It gives you back full control on everything. You can manage App permissions, you can fake permissions or if apps do not want to run with your set of permissions, you can even feed fake data (like wrong GPS signals, etc.). Read more here. http://repo.xposed.info/module/biz.bokhorst.xprivacy

4. F-Droid, yeah, the one open-source repository for your new apps. I'll install it at this point.

5. Now, that the device is flashed with CM, probably rooted and with a custom recovery, you have to flash a stock recovery again and lock the bootloader. Stock recovery because it does not allow any malicious party (hacker with physical access, police, intelligence services, etc.) to deploy any code to your phone which compromises your privacy. Locked bootloader is important to disallow any malicious party to boot anything they want which also compromises your privacy.

And this is pretty much what you need to get started, a rock solid environment free from Google. Make sure you have a strong PIN, I also use randomized screen locker, so people can not "observe" the way you enter your PIN.

For encrypted calls and SMS there is Signal, but that does not work without Google services and LibreSignal, the Websockets version, was discontinued just recently. For encrypted IM use ChatSecure rather than Conversations. Both are XMPP clients, but Conversations does not allow you to import or export OTR keys, which is very annoying for an Jabber client. For not so sensitive chats, I use telegram.

Finally, not having Google Play and Google Services available, makes the experience a totally different for the android device. Apps like Snapchat which do not require Google, but still do for some unknown reasons checks for Google, wont run. Also, a lot of apps work without Google, but you can't install them without downloading suspicious APKs from dubious websites. Be very carefull from where you download and install software if you can not find what you need in F-Droid.

I hope that helps you for your considerations.

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Edit, one more final note. I also use OsmAnd and have to say it never let me down on any occasion (except when I forgot to download the maps before going somewhere remote without internet). The geodata quality is excellent in most urban areas, but the interace and usability are a mess. If you find your way around in the interface, the navigation works out pretty well. I sometimes have issues calculating very long routes, but you start to live with that.