How to start developing Javascript apps?

(I realize my comments below typically refer to using javascript with regards to websites, so ymmv if you're using js for a different environment, like couchDB)

For an ide, any text editor will do, but I prefer convenient features like:

1. syntax coloring (because it's so pretty)

2. (s)ftp plugins - makes saving files convenient

I would probably go with something like notepad++, or aptana studio (although I find aptana to be a bit of overkill at times, but very powerful). Adobe Dreamweaver works nicely too, if you wanna pay. Otherwise, the bare minimum would be notepad and an ftp client (again, with regards to front-end web development work)

For debugging or trying out quick javascript, I find safari's and chrome's built-in development tools handy, but on firefox, it's firebug all the way. Firebug is awesome.

I notice that in my work environment that some people (particularly the business side) use the words javascript and the DOM interchangably, but I find that their attempts at explaining something simply leads to confusion. That being said, if you're writing web apps, I find the mozilla DOM reference to be pretty awesome and used to hit it up all the time before I got familiar: https://developer.mozilla.org/en/DOM

Actually developer.mozilla.org is overall pretty awesome.

Other than that, getting started is simply a matter of learning the syntax. I would probably get used to the syntax first (which should be pretty quick anyways) before I start thinking about other concepts like learning how to prototype and whatnot (which you can look up on this site).

A framework like jQuery is very convenient for various repetitive tasks, but I wouldn't necessarily jump straight into using jQuery without being familiar with the js syntax first.

The most important thing you should learn is jQuery. It is now the de facto standard javascript library.

jQuery is great regardless of what you are doing with it, but since you mentioned AJAX, I'll point out that jQuery makes AJAX stupid easy, because you can do things like this:

$.ajax({
    url: 'some/ajax/endpoint',
    success: function(data) {
        // do something with data
    }
});

Obviously there are a lot of other options you can pass along, but the basic structure is really simple and easy to use.

jQuery also has a really powerful syntax for selecting parts of the DOM and adding events. For example, if you wanted to catch anytime some clicked on an image and tell them the image src, you can do something like this:

$('img').click(function() {
    alert($(this).attr('src'));
});