Go to the Playstore (or Amazon Appstore). Search for "Overdrive". Download the Overdrive Media Console.
When it's on your Fire, go into the settings. To get there, it's the middle icon at the very top right. Looks like three vertical lines.
Put in your Adobe ID (e-mail address) and password. If you don't have one, get one using the button for getting one. Then put it in.
Go back to the library view.
Click on the symbol of a book at the very top right (left of the 3 icons).
Add a library.
Search for your library. Pick the right one out of the list. Click the star. Click the library.
Go to the library's Overdrive page (clicking the library will take you there.)
Search for a book.
Find one. Make sure it's Epub. If you get the Kindle version, you can still check it out, but you have to read it through your Fire and it's a few different steps.
Check it out. Download it.