I've always admired Colin Moock's books and I can recommend Essential ActionScript 3.0 as an excellent introduction to the language and paradigms of programming for Flash. I've also heard good things about Advanced ActionScript 3 with Design Patterns although I haven't read it myself. I do recommend being familiar with design patterns.
Adobe have moved towards Flex now when it comes to Flash for developers, so I recommend also becoming familiar with that. There is a free Flex SDK you can download and begin playing with immediately. Check out the Flex Developer Center for a ton on useful information about learning Flex. The core language of Flex is AS3 so it is transferable to standard Flash development, but Flex also includes a huge component framework and library and an XML based layout format (MXML). Flex Builder is a developer targeted IDE for building and debugging Flex based applications (or websites). Remember: developing Flex is free, only the IDE costs money (but it is worth it for the debugger IMO).