Inspect the DOM and set modification and event listener breakpoints:
Assets, Cookies, Databases:
Profile everything, network, scripts, styles, layout, painting, garbage collection:
* Safari has the same developer tools as chrome (they're part of the Webkit package), though Safari's tend to lag, and Chrome adds some stuff not present in mainline.
* Firefox has Firebug (third-party extension) and new releases have added a few built-in console-type stuff.
* Opera has Dragonfly, it's a built-in but (in my experience) tends to "feel" flakier than either WDT or Firebug.
* MSIE has its own devtools, they're horrendously bad in MSIE8 and still pretty bad in MSIE9 (they're flaky and tend to be unstable, they're also harder to use and extremely ugly). This is probably the devtools you'll find most painful. I do not know what will be in IE10. Purely for debugging I believe you can also hook Visual Studio (including Express Edition) and get a more full-featured debugger.
Some IDEs can also do remote JS debugging, which is actually pretty freaky. IntelliJ IDEA 9.0+ (and derivatives, RubyMine, PyCharm, PHPStorm and WebStorm) can be hooked in Firefox and Chrome with a bit of setup.
I also find the debugging experience better: the debugger seems stabler, and is definitely better:
* Default watches (scope variables) are more flexible (in fact, they're flexible at all)
* The stack pretty prints function arguments, you don't necessarily have to visit each level and check its locals
* Firebug hyperlinks objects and functions to (respectively) the object inspection tab and their source location
The WDT's timeline is not very useful in my opinion, the "network tab"'s overview on the other hand is great for understanding how resources are loaded and affect each other (or what the xhr sequences are when you seem to have a bug there)
> my debugger consists of alert()