Solving cross browser compatibility issues is quite a wide term. You might face different problems in Firefox, different in Safari, etc and that too for different reasons.
Some poses compatibility issues for CSS, some for HTML, you might face some compatibility issues for unsupported elements in specific browser and browser versions. There is a ton of things to be taken care of.
I have included a complete guide for cross browser compatibility in the form of links.
Hope this is all you’ll need to be an expert in cross browser issue solving and finding.