Besides the files that are easy enough to move over, for app data there really is no other choice than to either haves ones that support their own export/import functionality or if you're not lucky enough to have eliminated the apps that don't have it and need their data you can only go back to papa Google and ask to politely get all your stuff for restoration on the new phone.
Takeaways:

1. If you care more about salvaging data than privacy, use a Google account on your phones, otherwise, if you still value privacy but not so much security, root a phone as soon as you get it (not always possible or desirable) so you can use other backup solutions that require root access.
2. Prefer installing apps that have an embedded backup functionality so you can be sure it's always possible to get the data out regardless of what you did about point 1
3. (Bonus) Ask for said backup functionality to be added to apps you'd like to use with a feature request on the app's repo when it's open source, I've been doing that for the past year or so and I saw that quite a few have gone and implemented it, love these dudes :)
4. (super extra bonus) Fuck Google for artificially preventing a full backup solution that doesn't rely on their cloud being involved