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

NeoBackup is the only one I've run across that seems to really fill the role of backup and restore thoroughly. The trouble is, in order to work it needs root, so I've never actually been able to try it. Almost reason enough to root in my book 😅, I love a good back up system.

Seedvault is another fairly well developed option, but it needs to be hardcoded in to the OS by the ROM developer.

You'll probably benefit from a series of different backup apps in combination. Here's a few that I've used and benefited from:

SMS import/export - backs up all SMS, MMS, call logs and contacts. Does not backup RCS.

Applist backup - back up your installed app list. This includes data on where you installed the app from and where you can get it again along with other useful info. The apps still have to manually installed.

Aside from those two, most FOSS apps include a backup and restore function, such as: signal, neo launcher, fossify calendar, newpipe, metro (music player), aegis (2 factor), obtainium, etc...

I hope this helps. I tend to tinker and install various ROMs, so am well aquainted with the pain of setting up a fresh OS without a system wide backup program. Its not as bad as it seems though, and as long as you get your messages, contacts and call logs moved over it goes pretty smooth.

If you can't or choose not to use the Google provided backup solution, consider this https://www.openandroidbackup.me

Source code is here: https://github.com/mrrfv/open-android-backup

Companion app is available on F-Droid https://f-droid.org/packages/mrrfv.backup.companion/

Hope that's close to what you're looking for

A USB cable?

I use DAVx⁵ on my android devices, so calendar and contacts are synced to a private CalDAV/CardDAV server. (I self-host Radicale, but you could pay for such a service if you don't want to host your own.)

Pictures should be easy, since they're just files. Swapping microsd cards, copying to/from a computer via USB, file transfer app, KDE Connect... there are lots of options.

By messages, do you mean SMS/text? I suggest checking your old phone's messaging app for an export feature, and trying to import it on the new phone's messaging app. If that features is missing or incompatible between versions, maybe try one of these apps temporarily, just to get the backup/restore done:

https://f-droid.org/en/packages/com.simplemobiletools.smsmessenger/

https://f-droid.org/en/packages/org.fossify.messages/

Edit: More backup/restore options here:

https://search.f-droid.org/?q=backup

This is native android functionality.

Link

If it's really just pictures and contacts, do it manually. Contacts app should have an export function. Pictures you can just copy via USB.

Messages are usually cloud-stored nowadays right? Do you have more than idk 5 different messaging apps? If you don't, and one or some of them for some reason don't store messages on some server, find out if you can do it manually, most apps should have export/import functionality (not just messaging apps btw). For standard SMS though, no idea.