Buying ebooks on the Kindle store is pretty straightforward once you know the drill. You just need the Amazon app or to go to the website, find the book you want, and click the buy button. The payment method you have saved on your Amazon account gets charged, and the book gets delivered wirelessly to all your registered Kindle devices and apps instantly. It syncs your last page read, highlights, everything across devices, which is the main reason I stick with their ecosystem.
I did get tripped up once when I accidentally bought a book for a specific Kindle device instead of my account. Had to contact support to fix it, so now I double-check the 'Deliver to' dropdown. Also, watch for the 'Buy now with 1-Click' setting. It's convenient but dangerous for impulse buys! My recommendation is to always add books to a wishlist first and review it once a month. Saves money and makes you think about what you really want to read next.