First, make sure your Kindle is connected to your Amazon account. Set up an Amazon account online and save a credit card to the account. On your Kindle menu, type in your Amazon email and password. After you have accomplished this, you will not need to enter credit card details every time you buy a book.
Then press the Menu button and select "Shop in Kindle Store".
Browse through the books in the store, then click the "Buy" button to the right of the description, when you have found the title you want.
Wait for a screen to appear that says your book is being sent via Whispernet and will be available on your Home screen in a few moments.
Click on the book title on your Kindle's Home screen to begin reading your new Kindle book.