One thing that people often want to do when they've used React for a while is contribute to the library. React is open source, which means that its source code is public and anyone who's signed the Contributor License Agreement (CLA) can help to fix bugs, write documentation, or even add new features.
You can read the full terms of the CLA at the following URL: https://code.facebook.com/cla.
You need to make sure that any bug you post in React's GitHub repository is 100% replicable. Once you verify this, and if you want to file an issue on GitHub, you can go to https://github.com/facebook/react/issues/new.
As you'll see, the issue comes with some pre-filled instructions, with one of those being to set up the minimal demo. The other questions help you to explain the problem, and to describe current and expected behaviors.
It is important for you to read the Facebook Code of Conduct before participating or contributing to the repository at https://code.facebook.com/codeofconduct. The document lists good behaviors that are expected from all community members and that everyone should follow.
Once the issue is filed, you have to wait for one of the core contributors to examine it and tell you what they've decided to do with the bug. Depending on the severity of it, they might fix it, or ask you to fix it.
In the second case, you can fork the repository and write code to solve the problem. It is important to follow the coding style guides and write all the tests for the fix. It is also crucial that all the old tests pass to make sure the new code does not introduce regressions in the codebase.
When the fix is ready and all the tests are green, you can submit a pull request, and wait for the core team members to review it. They may decide to merge it, or ask you to make some changes.
If you did not find a bug but you still want to contribute to the project, you can look into the issues tagged with the good first issue label on GitHub: https://github.com/facebook/react/labels/good%20first%20issue.
This is a great way to start contributing and it is fantastic that the React team gives everyone, especially new contributors, the possibility of being part of the project.
If you find a good first bug issue that has not already been taken by someone, you can add a comment on the issue saying that you are interested in working on it. One of the core members will get in touch with you. Make sure to discuss your approach, and the path you want to take with them before starting coding, so that you do not have to rewrite the code multiple times.
Another way of improving React is by adding new features. It is important to say that the React team has a plan to follow, and the main features are designed and decided by the core members.
If you are interested in knowing the next steps that the library will take, you can find some of them under the label Type: Big Picture on GitHub: https://github.com/facebook/react/labels/Type%3A%20Big%20Picture.
That said, if you have some good ideas about features that should be added to the library, the first thing to do is open an issue, and start talking with the React team. You should avoid spending time writing code and submitting a pull request before asking them, because the feature you have in mind might not fit into their plans, or might conflict with other functionalities they are working on.