I've got the process down to a science. First, make sure you have the Kindle app installed on your device and that you're signed in with your Amazon account. Open the Amazon website or app, search for the book you want, and look for the Kindle edition. Once you select it, hit the 'Buy now with 1-Click' button if you have that set up, or proceed to checkout normally. The book will automatically appear in your Kindle library within minutes.
If you're using a non-Kindle device like a phone or tablet, make sure your device is linked to your Amazon account in the Kindle app settings. Sometimes, you might need to manually sync your library by pulling down on the screen. Also, keep an eye out for Kindle Unlimited or Prime Reading options if you're a subscriber—those let you borrow books instead of buying them outright. It's a seamless process once you get the hang of it, and I love how quickly new books appear on my device.