Yes, Firefox is the new IE, literally. In the past years generally everything just works, i develop in Safari and then test in IE, Opera and Chrome. Usually no big quirks. And then comes Firefox, usually something is badly broken. Let's say i have added extra tags into DOM just for FF only, I have FF specific CSS hacks.
The first thing you learn as a web developer with respect to (modern - ignoring IE 7 and even 8) cross browser compatibility is that all browsers have their quirks and all of them suck. Off the top of my head:
Until recently, Chrome 'implemented' the HTML5 date input by giving the user a spinner on the side of the input box which, when pressed, will set the date in the input to '00-00-0000'. Worse, this renders the normal feature detection used by libraries such as modernizr.js useless.
There was a bug that caused inset box shadow used to be rendered outside the element. This is fixed in recent desktop versions, but still haunts older versions of Android's default browser - versions as recent as the one included with 3.x Honeycomb are affected.
Browsers, even modern ones, are incredibly quirky. Moving to a monolithic Webkit market will not help with any of the bugs listed above, nor, I suspect, most of the bugs one might encounter today as a web developer.
The first thing? Well, I have been web developer since 1999. I wrote drag&drop cross-browser JS in 2003. Under my belt is front-end of one Seedcamp winner, webapp that was presumed as native on iOS. What exactly do you wan't to teach me?