Where is the best place to start learning cross-browser compatibility?

Question

css

Hi friends.
I'm learning frontend. I am already well versed in (adaptive) layout, but the question arose of how to adapt sites for different browsers. Of course, there is a theory of progressive improvement, etc., but I would like to hear some clearer recommendations, because. There is a lot of information for adaptation to different types of devices, and all this is mainly limited to grids + media queries, but there is not enough fresh information about cross-browser compatibility.
Perhaps you will advise some relevant books or libraries that fix some problems with displaying the site on older versions? Is it worth chasing IE hard now?
Thank you.
UPD. How much attention should be paid to older browsers? Those. how much further (from the latest versions) does it need to support older browsers?