I worked 13 years for a QA organization. hardware and software. Not games, but I have some insight into that industry as well.
Different organizations have different ideas about the kind of people they hire for testing. But it all still comes down to using features, indentifying spots a bug could occur, trying it, documenting it, reproducing it, and later retesting that the bug was fixed.
Generally, you want people who are detail oriented, because you need to be able to spot sublt bugs, not just blaringly obvious crashes that anybody who presses the Start button could find.
A lot of times this means trying the same thing over and over again, with subtle variation.
It'll mean keeping good notes and writing up a detailed description. How you write a bug report can determine how serious it seems. Furthermore, if the developer can't reproduce it, he'll CND it (Can Not Duplicate) and it'll be closed. You don't want too many bogus bugs in your name in the bug tracking tool.
I started in QA, doing testing. I did manage to move out of it into development, where my degree was. Most folks don't. Where I was at, they hired all college grads. That's expensive. The trend is to hire leads as college grads and give them contractors (non-graduates aka cheap labor) to do the testing for them while they wrote test plans and oversaw the work.
In a game company, your chances of getting out of QA into something better is less, especially if you're not degreed.
Consider the stereotypical game tester: no college degree, minimal if any formal programming experience. If you're working fulltime, your not going to learn to program on the job.
A friend of mine apppied for a programming job at EA. They sent him a take home test first. It was involving complex mathematics for 3d rendering. The short of it was, a guy who knew how to code for databases, drivers, and firmware couldn't do it. It takes a certain sub-set of programmer to do graphics programming.
Game programming requires very optimized for speed code. I had one developer on the team from Broderbund. Her coding habits were geared for that, which made for unreadable code.
Generally, game designers come from the ranks of programmers, or level designers. People who have experience in making the games generally know how to make more games.
If you want to get into that, go check out XNA which is the free dev-kit for writing xbox games (and windows games).
In general, I wouldn't hire a kid to design a game for me if he didn't have a portfolio of games in the genre I wanted. for video games, that means you need to make software. the tools are free. So anybody who doesn't do it is just a wannabe, and not serious.