Hi John!
Right now, the "React Core" team at Facebook is about about 5 people. The React core team is responsible for the web version – React DOM – as well as all the bits that are shared by React DOM and React Native. (There are separate teams that focus on React Native.) We also take care of a few other projects like create-react-app and react-devtools.
Each day can vary quite a bit depending on the person (and day), but it usually involves some combination of responding to GitHub issues, fixing bugs, and building new features for React that we think will help people build great apps. We also help other Facebook engineers when they have React questions and handle updating Facebook code when we introduce new deprecations, but the vast majority of our work is done in the open on the GitHub project so you can usually see what we're up to. New commits to React always land on GitHub first, then we import changes into our internal repos once every week or two.