Move data (without cloud storage) from CM 13.0 to LineageOS My current phone is running CM 13.0. I have a new phone ready, running LineageOS. What's the simplest way to move all data (most important: contacts, pictures, mail + configuration, SMS) to the new phone?

Both phones do not have Google services installed and I don't want to move my data to the cloud. I tried to make a adb backup on the one phone and a restore on the other phone. I did not run into an error, but I also did not see the data on the new phone. I would be happy to pay a small fee for a decent tool, but without access to the play store, most apps are not an option.
First and foremost: provided that you trust 3rd-party APK provider sites (APKPure, APKMirror, etc.), no Play Store access is not a problem - you will only be restricted from paid apps. For trust issues read this question (and others related) on the site.

Contacts:


In the Contacts app, top-right menu - import/export - export to .vcf file.
Select items you want exported.
A .vcf file will be saved by default to the root of the internal storage - back it up somewhere.
Import later with the import from .vcf file option.


SMS: use the app SMS Backup and Restore. Been using the Pro version of the app since Android 4.x and it always worked smoothly. Backed-up content will be stored in the SMSBackupRestore folder under internal storage. APKMirror

Mail: your mail is supposed to be all up in the clouds, i.e. synched with your mail server, so there should be no need of backing them up in the first place.

Media files (picture, music, etc.): provided that you know the location, just copy them off the device. Use the details option in the Gallery app if you need help locating a specific picture.

App data: the free version of Titanium Backup is enough for backing up individual apps' data. Backed-up content will be stored in the TitaniumBackup folder under internal storage. Note that, before restoring app data, you might have to start them once. APKMirror

Copy backed-up contents off the device (keep folder structures intact), then after flashing LOS, copy them back in place, and use respective restore options.