The Kindle app does not allow you to buy/download books within itself. The way it is designed you have to go to the Amazon site through Safari to do this.
Actually, from the Kindle app, Home screen, you cam tap on the "Shop in Kindle Store" on the upper right of the application. Doing this takes you directly to the Kindle section of Amazon. You are using Safari, but you do not have to open Safari. It is very simple and effortless. You must have either WiFi or 3G operational before this will work.
Gordon, as others have said you have to purchase the ebooks at the amazon site through Safari (whether you do this directly or through the app makes no difference).
After you have purchased a book, then you go into the app itself. At the lower left corner it says "Home". This screen shows the ebooks that have been downloaded. If you tap on it it gives the option of "Archived" which is where books are stored that you are not currently reading.
Also, you have to register the Kindle app so that your online account at Amazon.com knows to send any new purchases to your iPad. That way when you next start the app it will automatically download your new purchases.
Hope this helps because the app, while not as simple as iBooks to purchase ebooks, works fine.
