I'm currently using a Nook, but considering switching to Kindle hardware. Is there a simple way to keep books synced across devices, or at least to move purchased ebooks from one to the other?
The most effective way to manage your Nook books on your Kindle (without removing the DRM) is to obtain the Nook for Android app and sideload it on to the Kindle (you'll need a Fire or later model to do this).
If you don't have another Android device handy, you will need to find a means to download the Nook APK file to your computer.
You may want to try using the Calibre Ebook Manager with the deDRM plugin to get the ebook files, sans-DRM, and then move them onto the Kindle. It works, but it won't sync them or automatically move newly purchased books; each book must be done individually. The Digital Reader has some helpful tutorials on how to do this:
Note that DRM removal is illegal in some countries. Please read and follow your local copyright laws.