"Game Developer" is a misnomer. There are
many jobs to be done in the "game development" arena, and most are not development of the game. Just to name a few, you have:
>> tools devs, who build the tools needed to manage the game environment;
>> server devs, for those games with a multiplayer component'
>> network stack devs, because that piece of code must be bulletproof and FFFFFAAAASSTTT!; and
>> AI devs, to guide the NPC's to their prey...;
Gone are the days of Epic's Sweeney and ID's Carmack, guys who would sit and write an engine in a week. Valve-type games are now million-dollar budget epic sagas. That said, there's nothing wrong with Plants Vs. Zombies... or Angry Birds... they've both made millions.
You need to decide what parts of the process appeal to you, and focus on those. Or, you could write a game, from the ground up, and spend a few months to a year cutting your teeth to learn how it works.
Then, there's always an internship if you feel you can afford a few months working for nothing to get your foot in the door.
If it were ME, and it has been (I sold my game company back in '94 for a pittance), I'd spec out a simple game, write it, add a network component, write it, and get some friends to test it in play to see how it goes. You don't even need cool graphics to start... those can be added later if you design it correctly.
Good Luck!