Hardware-wise, you will need a Mac for this book. The content in this book is iOS-first, and to develop iOS apps you must have an Apple computer. Optionally, both an iOS and Android device would be helpful for testing apps on-device, but not mandatory. There is one API in the final chapter of the book that requires a physical device to test (Vibration), and another where testing would be easier on a physical device (Linking).
You will need to install the React Native SDK for your Mac. Instructions can be found at https://facebook.github.io/react-native/docs/getting-started.html. Prerequisites for installing the React Native SDK are available on that page.
Instructions to install Xcode and Android Studio are also available on that same page for installing the React Native SDK to your machine.