Start with Firebug, IE Debugger.
Be careful with debuggers in Javascript though. Every once in a while they will affect the environment just enough to cause some of the errors you are trying to debug.
EDIT
One comment requested examples:
For IE it's generally a gradual slowdown, some kind of memory leak type deal. After a half hour or so I need to restart. Seems to be fairly regular.
For Firebug, It's probably been more than a year so it may have been an older version. As a result, I don't remember the specifics, but basically the code was not running correctly and after trying to debug it for a while I disabled firebug and the code worked fine.