CSS Modules are one of the most popular ways for styling React components. Whether you are using only CSS or a more advanced pre-processor like SASS, it doesn't matter for CSS Modules: You can write all these styles in your style sheet files next to your React components.
Fair warning! This isn't a speedy, straightforward, expert-driven plow-through of how to set up these technologies. Although, by the end, we do successfully get it all going. This is about documenting the real-world experience of doing this kind of work. Some things work easily, some don't.