There's no need to spend a lot of money on e-books for your e-reader. Most libraries now lend e-books for a variety of e-readers. I've boiled down the process for borrowing a library e-book for any Amazon Kindle e-reader device such as the Fire or Kindle app to six easier-than-they-look steps.
Or, make sure the card you have is active - and you may have to visit your library in person to either get or review a card. In either event, modern library cards are likely similar to this one from the New York Public Library. On the rear is a bar code and a library card number. During the library card procurement process you'll be asked to create a PIN or password, or you may be able to sign up for a pin on your library's Web site. You'll be prompted to sign-in to your library's Web site using this card number/PIN and perhaps during other stages of the e-book library process.