I get asked for career advice all the time. I sent this to a university teacher recently, but it applies broadly. This same message hasn’t really changed over the years.
Decide what you want to do all day. Programming, 3d art, audio, map layout, etc. Pick one. Get even more specific: Programming AI, building multiplayer action game maps, etc. Become great at that thing…so that managers want you on their team.
The idea is to become trusted and autonomous; a fire-and-forget team member, where people say, “If we put him on it, he’ll make the right calls, autonomously, and it’ll get done, and it’ll be solid.”
Be as technical as possible. Even if you have a focus on story, human interface, mission planning, or music, make sure you’re as technical as possible; familiar with editing tools and software.
Be well-versed and well-rounded: Know not only your area (like software engineering), but be familiar with media theories, interface, psychology, sociology, various gaming platforms, social trends among gamers, general design, etc.
Look over the commonly recommended game design books: Rules of Play, Hamlet on the Holodeck, Understanding Comics, etc.
I love it when someone comes in, has passion for game design, is a great collaborator and communicator, can talk about Malcolm Gladwell or Scott McCloud, has read the Design of Every Day Things, has played a ton of games on many platforms, can program in C++ (or use Maya, 3D Max, UnrealEd, Source, et al), has written some fiction, can teach/lecture, has travelled, and is both strong-willed and willing to change her mind.