You purchase books by going to the
Amazon Kindle store in the browser (Safari), or on your computer. Buy them there. When buying you have an option (usually) to have them delivered to your device. If you haven't registered the iPad app yet, it won't appear. Not to worry. You can download the book directly from the iPad later.
On the iPad, you need to open the Kindle app and register (sign in) with your
Amazon account. It will probably prompt you when you open it, but if it doesn't you can find the option by taping the 'i' icon at the lower right of the screen, then Settings.
Once you are signed in (and have an internet connection) you should see all the Kindle books you've purchased on the Archive page. Tap one to download it and start reading.
To make your life easier in the future, once you've navigated to the Kindle Book store at
Amazon.com, tap the box-and-arrow icon just to the left of the URL bar, then choose Add to Home Screen. This will create an icon on your iPad that is a shortcut to the Kindle store. You can put it right next to the Kindle app as a reminder.