Debugging a facebook application is practically no different from any web application:
Client side:
Server side:
- Firebug
Net and FirePHP tab (if you are comfortable using it) - main
var_dump and echo
What is really hard for me to understand is their documentation!
Extra tip :
Try playing with FQL methods by creating a test group / page and events ... to better understand how the FB Graph API works. This will make reading documentation easier.