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.
After you click "Get for Kindle," you'll be transported to an Amazon Web page on which you can choose which Kindle device to send your borrowed e-book (red arrow). This list includes not only Kindle devices, but any Kindle apps you may have on a smart phone or tablet (yes, I have a lot of registered Kindles - it's what happens when you review a lot of smart phones). Choose one Kindle device initially. You can go back into Amazon's "Manage Your Kindle" account page and transmit your borrowed book to other Kindle devices or apps. After you've chosen your device, click the "Get library book" (blue arrow) button.