You'll definitely need the Kindle app, it's the easiest path. You can get it on your phone, tablet, or computer, and once you've purchased the book from Amazon, you just tap the download button. It's a simple toggle. Make sure your device has enough storage, but ebooks are tiny so that's rarely an issue. I do this all the time before flights.
Actually, a common hiccup is trying to do it through a web browser. The download only works reliably in the dedicated app. Open the app, go to your library, find 'Cat's Cradle', and there should be a little cloud icon with a down arrow. Tap that, and it's yours. The file gets stored locally, and you can open it anytime, anywhere. I love that feeling of having a whole library in my pocket, totally disconnected.