How to back up phone without Google?

Replace the stock dialer, contact and sms apps (and some others) with Simple Mobile Tools. They have the ability to export/import, and can use a separate contact list invisible to all other apps or the system.

For the files, Syncthing is a good choice.

KDE Connect can also do a lot of things you may want.

@Stach There plenty of solutions depending of the problem you have :
You can backup your data on you PC with tools like syncthing.
You can also self-host a backup solution on a private server and use tools made for this.

for the second solution, you can use :
Etesync for your contacts/calendar/tasks
Immich (https://immich.app) for your photos/videos.

If you juste want to backup your phone with data/apps. It depends of the ROM you're using. I use LineageOS and Seedvault for the backup.

And for your documents/other files, You can use a cloud storage respecting your privacy like proton drive.

What OS are you using on PC?

If you are on Windows: https://www.fjsoft.at/en/

I use the integrated /e/OS backup which is actually Seedvault which runs great :). But you need a service to store you backups, WebDAV, SMB or etc. Neo Backup (which I have used for many years on Lineage OS) seems to be able to make backups to a SD card.

I use a Synology NAS to back up and sync everything. There's a high up-front cost with this option, but there are a lot of benefits and they have great apps for accessing stuff. The argument can definitely be made that it's more cost effective to just pay for a secure cloud service subscription, but it's nice to self host, plus now I have 16 terabytes of space and can run my own services like bitwarden and Plex for zero cost!

You can easily export contacts as a file and import it in new device.

Files can be easily copied as well.

SMS can be backed up and restored by Super Backup app (or other similar apps). I haven't tried it though, and it might not do it perfectly.

https://syncthing.net/ could handle the transfer of the files to your pc but i'm not sure how you'd get it to do your sms