Avoiding clouds is easy: there are apps that will write a backup to your phone's internal storage, which you can then copy to your PC over USB. The venerable SMS Backup and Restore is an obvious one (it has a cloud option, but you don't have to use it). If you want to avoid Google, look in the F-Droid store, there are a couple of options there that do the same thing.
The message database lives in the part of the phone's storage that isn't accessible over USB, so as Danny says there is no way of doing this without a third party app that makes a backup and puts it where your PC can see it. In principle you could copy the database over USB using ADB (Android Debug Bridge, i.e. the USB debugging interface), but you'd need a rooted device for this to work. Also I would not expect that method to allow you to transfer the messages to a new phone: for one thing the new phone would have to be rooted first, but also the database isn't designed to be portable between devices and there is no guarantee that it would work if copied to a different model running a different OS version (in fact, based on my experience when I used to root and modify phones, I'd think it highly unlikely that it would).
So you need to use a third-party app, but there are open source options that don't require any clouds or even using the Play Store. That really should be sufficient (if it isn't I don't know how you can use a Chinese phone whose operating system is controlled by Google in the first place
).
