Personally, I've gone down several different paths but they were all tied together by a single overarching theme. I like to make worlds, and I like to give them life. Worlds in which others can be immersed, worlds isolated from our own, worlds to which people can escape. I learned how to draw and illustrate because it was a way to bring them into existence. I learned to program because it was a way to give them structure and make them real. It's this reasoning that will determine your path, which is why "I want to make a game" is not enough. Wanting to do something just for the doing of it will leave you rudderless in a sea of options. Wanting to do it for the glamour of success will leave you drowned once your boat's been punched full of holes by the failures we all must face.
We can't always put the reasoning to words at first, but we know it's there. Part of the journey is giving this shapeless motivation form and substance.
Anyway, enough of this rambling about the abstract. I mentioned initially that I'll share my experiences with a part of the industry, rather than the whole thing. There's so many different studios out there, and depending on their scale, their teams can be structured in vastly different ways. The usual trend is that in larger studios (think triple A), people are hired to perform a specific task. Everyone's got their own specific skill set, and they stick to it. Specialists thrive in this kind of environment, and it can be an exceptional learning experience, as there's usually lots of opportunities to learn from your seniors. Smaller studios (think indie) are a whole different animal. If you've got skills outside of your official position, they'll be put to use in one way or another. The way my boss puts it, people are encouraged to wear many hats. It's not just limited to the skills you arrive with. If you're interested in expanding that skillset, it's heavily encouraged.