How to start programming Flash under Linux, using Open Source tools where possible?

That is mainly, programming Flex. Which is a framework that mainly consists of a runtime library on top of the basic Flash runtime library. Helpful tools:

• Flex 3 Language Reference.
• Flex SDK Documentation overview.
• Tutorial for starting programming in ActionScript 3.0.
• All Flex documentation in a single ZIP.
• IDE alternatives (by appropriateness)
  • Flex Builder, an Eclipse-based Flex IDE from Adobe, commercial, free Linux Alpha. There is a gratis alpha version available for Linux. It has a support forum. But it seems that Adobe will never release Flex builder for Linux officially [source]. The application needs Eclipse 3.3. If installing it in Eclipse 3.4, everything will work except for the MXML editor. But there are approaches that should work for Eclipse 3.4 installations [source, esp. this link there and this] and Eclipse 3.5 installations [source].
  • AXDT, an Eclipse plug-in and the best open source IDE, also in active development.
  • IntelliJ (or its 30-day-trial version) is a very good choice, but commercial.
  • Powerflasher Solutions FDT, commercial and expensive, free 30-day trial. See here. It is an Eclipse-based IDE. Installation on Linux is possible, while there is no official support.
  • Eclipse Colorer plug-in version 0.8.0, available from SourceForge. Open source but just a syntax highlighter.
  • FlashDevelop, available from its website. Open source, popular, but Windows only.
• At least Flex Builder for Linux and AXDT do not have a code formatter. This can be added to Eclipse as another plugin. The project is named Flex formatter. Install by adding the following update site to your Eclipse installation: http://flexformatter.googlecode.com/svn/trunk/FlexFormatter/FlexPrettyPrintCommandUpdateSite/.
• Depending on the IDE you use, you will have to install the Flex SDK yourself if the IDE did not come with it. For that, download the free Flex SDK via. More bundles and versions are available from here. The Flex SDK is not (yet) in the Ubuntu 9.04 (Jaunty) repositories. To install a Flex SDK, unpack it to a convenient place below /usr/local, and symlink all files in the bin/ subdirectory to /usr/local/bin with the following commands:
  cd /usr/local/bin;
for file in ../share/applications/flexsdk_3.3/bin/*;
  do ln -s $file .;
done;
• Some IDEs like AXDT can also work with Adobe’s Flex SDK 4. This is however not advisable to install, since it’s currently just a preview release.
• More helpful tools:
  • Flair: an Eclipse plugin to view SWFs and their log output. However, when using AXDT, there is already an internal SWF / SWF log viewer. Download from here.
  • Eclipse plugin for viewing the Macromedia’s Flash MX Professional 2004 LiveDocs within Eclipse. Download from here.
  • Apache Ant Eclipse plugin, for build automation.
  • See also this list of more open source flash tools and projects.