One of the unique things about the new Kindle Fire is that it's a dedicated Amazon eReader that runs on Android. If you've got a Kindle e-Ink reader, it's a little harder to read Nook, Kobo, or Google eBooks. Part of the problem is that most readers, like the Nook reader, use ePub format. Amazon uses its own .mobi format as well as Adobe PDFs, but it doesn't read ePub books. You can convert your Nook and Kobo books by using Calibre, but that's a bit of a pain if you want to keep your books in sync on all your other devices, like your phone or other eReaders.
Because the Fire runs a version of Android, you can actually install and run the Nook or Kobo app. You can't download those apps from the Amazon App Store, but you can still install them by sideloading the apps instead of just sideloading the books.
Your Nook and Kobo books will not show up in the Kindle Fire carousel. Only the app will show up, but you can see all your Nook and Kobo books within their respective apps, and you can make in-app purchases to buy new books.
This method will work for installing just about any free app you can't find in the Amazon App Store.