There is currently only one way of getting a job as a games programmer, and that is to write a game - or more properly, write what is known in the industry as a game demo. These days, a typical commercial game might take 100,000 lines of C++ code written by a team of 3 programmers over a period of 18 months with a budget of a million dollars. As a novice game programmer, you are obviously not going to be able to compete in that arena. But you can and should write a fully functional game of about 10,000 lines of C++ code that has at least one and preferably several features not found in commercial games. Game companies are less interested in your resume than in what you have done. So write a game, put it on a floppy, and send it to the game companies that you would like to work for. If they like it, you may strike it lucky.
So where are you going to get the knowledge to write your game demo? One way is to get a college degree in Computer Science. Do you have to have a college degree to get a job programming games? Not necessarily, but things are changing as the industry matures. You can get a job by programming a drop-dead game demo, but many companies are looking for employees who have produced great games and have college degrees. This makes sense because college is a place where you can get the background knowledge that you need to be a competitive game programmer. If you are fortunate, it can also be a place to meet design people who can join and support you in your quest.