First, you have to have the iPod (and its Kindle app) registered on an Amazon account. Then, the only way you can get books to the iPod is to go through a regular browser and search in Amazon to find any free books. You can't download books from within the app.
Once you find a book you like at Amazon, there will be a button on the web site so you can choose to send the book to your iPod (that's why it needs to be registered first). Amazon will send the book and then you can open the Kindle app, select the book and start reading. By the way, this works for both free and paid books.
Hope this helps. Do let us know how you get on.
Marilyn