What are some resources for getting started writing a Firefox Addon? Is there an API guide somewhere? Is there a getting started tutorial somewhere? Is there a developer discussion board somewhere?
This is a great resource to start learning how to build a FireFox extension:
How to create Firefox extensions
This is an awesome tutorial and will covers most type of extensions.
Here's the official starter page from Mozilla for writing your first extension. http://developer.mozilla.org/en/Building_an_Extension
We tried to make http://developer.mozilla.org/en/Extensions answer all those questions. The first three links in the documentation section are about getting started (that includes Adam's link). The newsgroup and the irc channel in the Community section are the official discussion boards.
Mozilla is very complex, so any kind of API guide would be overwhelming and hard to write. So your best bet is to check the code snippets page (also linked from the MDC Extensions page), then search MDC/google, then ask in the forums.
Consider signing your finished extension to avoid the untrusted extension warnings.
http://o-regan.org/2007/04/14/xpisigner-java-cross-platorm-xpi-signing-tool/
I found greasemonkey to be a great starting point... I used it to create some functionality for a site, then I used this script compiler to turn my script into a working add-on. Of course it's machine generated... but it's very few files and pretty easy to understand. Just unzip the .xpi and tinker away.