I recommend using Webpack to bundle your code. It will roll your whole app together into one file (or a few if you get into optimizing webpack a bit more). You can then have a very basic index.html file that that simply loads the bundled js file. Push that to your production server in whatever way you like.
Here is a good tutorial to get you started with Webpack: Setting up React for ES6 with Webpack and Babel (there are a lot of these out there if you don't like this one)
One of the current challenges with React is larger-than-optimal bundle sizes. For complex apps this can start to be a problem for initial page load. Enter isomorphic rendering. React can run server-side and render a snapshot of your app that downloads quickly. Then when your actual app bundle downloads it seamlessly picks up with the current DOM, making for a much snappier user experience when they arrive on your page.
Here is an example for doing isomorphic rendering with React: Example on Github
Go throw "ReactJS isomorphic rendering" in Google for much more info.
Good luck and have fun - this is cool stuff :)
