For me it was quite the opposite. When I first started programming at age 12 my goal was to learn as much as I could about animation and game programming. Then when it came time to get a job I went another way and became a web developer.
Perhaps I'm biased, but I do think game programming is the most fun I've had writing code and is a great way to learn new concepts. It presents interesting challenges too. There are all sorts of computer science concepts that must be applied as you get into more & more advanced games, animations, and effects.
And then there's one aspect of game programming that I think we can all relate to on some level. And that's the necessity to optimize the crap out of everything to maximize your framerate. That, as it turns out, was probably the single most important skill I learned from it. That kind of optimization knowhow can be taken and applied to all kinds of programming. And it's still to this day a valuable skill to have in my day to day job.
Speaking of optimization, game programming can be done in all sorts of languages. But I'd suggest Visual C++. It's still pretty much the defacto standard for most games developed today because of its speed. And there will be plenty of C++ examples out there done in Direct3D (Windows) and OpenGL (cross-platform). I went the OpenGL route and went on to create gldomain.com when I was in high school, but it's your choice.