How to get going with AS3 for $0 a day, on Windows, Ubuntu, or OS X
At the simplest, flash development just requires the (free) Flex SDK from Adobe. The current version is 4.x. I will mention a few relevant differences between Flex 3 and Flex 4 as we go. Depending on your bent, you can use the fully open-source version, or the "free as in beer, not freedom" partially closed-source version. I've always used the "free as in beer" version. For our purposes, the Flash SDK requires Java 1.6+ to be installed.
You will also want the Adobe-supplied debuggers. You can get along with just grabbing one, but I always get the set (ActiveX, plugin, and projector).
Okay, things to download:
I strongly recommend going the FlashDevelop route. This will wrap the Flex SDK for you in a delightul, intellisense-style IDE. Go to the FlashDevelop homepage, note the "LATEST RELEASE" box on the right side, and select "Release notes and download". As of 2011-05-26, the latest release is 3.3.4, if that matters any. I've been using FlashDevelop for 1.5 years, and have always upgraded painlessly.
Else, just follow the Mac OSX/Linux directions. ;)
After unzipping the flex sdk, you'll want to put it somewhere, and then put its "bin" directory in your path. You'll be doing your development from a terminal or shell. So to make sure it's all good, type "mxmlc". If it says "command not found", you're not ready.
Let's move on to the first tutorial and see if you've got it.