I second Conagher's suggestion of Code Academy[www.codecademy.com] and would also like to suggest Code School[www.codeschool.com]. They do a great job of "gamifying" learning to program and have some quality, easy to follow tutorial videos that go along with every lesson. You can pay money to get the full experience with Code School[www.codeschool.com] but i know there are free lessons as well.

Colin,
If you haven't already, I would take a look at the valve employee handbook[newcdn.flamehaus.com].
I think this succinctly describes the "mastery" concept that you are looking for regarding game development (which in and of itself is a very broad category). In short, it says be a specialist at one area and have knowledge in other areas. I'm sure many of the members of the pipeline team would attest to gaining their internship position through this sort of specialization. Also, to echo what Sam Za Nemesis mentioned, working in a team will be your best bet. With the breadth of knowledge out there it is next to impossible to do it all by yourself successfully. I say all this not to discourage you but to encourage you to find your area of greatest interest. Good luck on finding your area of talent and pursuing that with everything you've got. As a cross country/distance runner, I like this quote from Steve Prefontaine[en.wikipedia.org] "To give anything less than your best is to sacrifice the gift."

-Kyle "LegendarySpork" Boe