So we already posted a bit here and there and maybe tried to use some code blocks, there is the "Format" - "Code" option but it looks garbage when trying to provide actual code
(WordPress autoformats a LOT, and I already deactivated everything I could)

So this is how we've been doing it:

Looks kind of ok, but is it really that helpful for actual code in a post?
Absolutely no.

A new button appeared in your editor, which provides you following:

Looks way better and is much more helpful for code! Will become handy, for sure.

The editor comes with certain modes, I'd always keep "Generic Highlighting" as Language, but you can choose between two modes, depending on your use:

Block Code is the example above (the code expands the whole width of the post), but Inline is pretty useful for in-text-code, like: