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It does kind of depend on what you mean by building quickly. The best advice I can give is to build an iframe application. This lets you run your own internal server, and that means you can track analytics events, so you can build small fast, and then test and iterate quickly as you go based on hard data.
As for the facebook integration itself, their latest API documentation is at http://graph.facebook.com/ . You'l have to register your app with them at http://facebook.com/developers . Let me know if you need any more details!
you need facebook library, you can download from google or facebook.
if you know action script and html then have look at http://code.google.com/p/facebook-actionscript-api/.
you can use Flash or Flex to make facebook apps easily.
you can look at my example here: http://apps.facebook.com/Classic_Hangman/