If you are concerned about Google, then you might as well rule out most other cloud services as well. They are all guilty of compromising your privacy in the name of convenience. Which means you'll have to back up your phone to a PC or personal server, and you'll have to do it individually.
The easy part is copying files and folders from your user partition. Just plug in the phone to any PC, enable MTP mode if it's not enabled already, and copy the entire contents of the /sdcard partition to a folder on the PC. Trickier are things like email and text messages, which are stored in a protected database on your phone. There are several
decent apps to backup SMS and MMS messages to a single file that you can then restore on any device. Email will depend how it's configured. If it's IMAP, then those messages are on the server and you don't need to do anything (unless you want to keep your "sent" folder, too and that may or may not be synced with the server.) If it's POP3, then the easiest way will be to simply forward them to a different email account.
The problem now arises with app -- especially paid apps -- that save data to your phone. A lot of times simply copying the files to your PC won't reconnect to the app when you copy them to a new device. And, you won't even have access to paid app folders since those are stored in the system partition. And your settings like WiFi passwords will all need to be reset, too.
Maybe if you list what's a on your "must keep" list, we could be more specific.
