How to perform the implementation of tests?

I currently have an application with my backend in PHP (Codeigniter) and Frontend in Vuejs. I know that the test culture is super important to ensure the quality of a software, but I have a lot of doubts on how to start:

Where do I start? Do I write the tests for the backend or frontend first?

Before deploying the application, should I run all tests? Backend and frontend?

I know that there are test frameworks for PHP (PHPUnit) and for Vue as well (Jest) I wonder if there is any way to test the styling part, too, of CSS. Because sometimes i often create a class in my style sheet that breaks some other element of the application.