Javascript, like html, is all client side so its easy enough to code in with a simple text editor and refresh. Chrome is now my default browser so I use the js console for all my debugging. I just code changes, hit refresh and watch the console. No errors generally mean the app works as intended. Syntax errors bring up a red x and the line number. The console is also very useful for debugging and verifying asynchronous requests. Logging to the console is also very helpful using console.log();. Lastly the debugger helps you track down 404 requests like old paths to images and does general asset speed optimization.
A common example would be attaching handlers to some links via jquery. I usually start with asserting that a jquery click handle is executing. I would console.log('works'); and refresh the page. If nothing logs, then I look if any errors were reporting, has the jquery file loaded (the console tells you all this).
Second thing I use a lot if ajax so I have to know if I'm getting a response back, and what the response is. A lot of times the response can cause a 500 error and you won't see any updates from the browser side. Using the console you can see whether or not the request was made, check headers, and response output. An error response is conveniently reported with bold red dot to indicate a problem.