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.
Here are three consecutive screens to confirm your lending choices. Once you find a book for which an e-book edition is available, click the "Add to eList" button (red arrow). On the subsequent screen, you may be able to choose the length of the lending period (blue arrow). Once your 7, 14 or 21 days are up, the book will disappear from your Kindle. Make a note of your due date - I didn't see a counter on my Kindle warning me how many days remained on my loan. On the next screen, click "Get for Kindle" (green arrow). Note: your library's Web site may not look or act like these examples exactly, but it will likely be close.