The first thing you need to realize is that no one gets a design job first thing. There are a lot of other gigs in the industry other than design and programming, which is something most people aren't aware of at all! To be a designer, you have to know a lot more about the gaming industry beyond the games you like to play. That means that step one is finding out more about the process of getting a game from conception to market.
You'll want to look at industry magazines and blogs to read up on what's happening in game design. Keep a close eye on all the games that are coming out (not just the ones you love to play) and try to get in contact with a few people in the industry. That doesn't mean sending email to your favorite designers—they're probably swamped. However, there's a pretty good chance you'll meet someone with a game-related job who can tell you more about the field. Remember to learn about multiple platforms and genres, since you never know where you'll end up.
Next, you'll want to think about your skills. You won't just get to sit and think up ideas that other people put into practice. You're going to have to be able to create a game design that's feasible. That means having good writing ability in both the creative and technical areas, knowing design theory, having a good grasp of computers, knowing about code, and having some basic understanding of computer art. You should also be able to sketch and storyboard and know how to work with others, since almost all games are a team effort.
You also have to have experience. The best way to get into the games industry is by doing an internship. They're hard to find and competition is stiff, and you may not be able to find one that pays at all. Those warnings aside, an internship will get you into a company so you can see how everything actually works on a day-to-day basis. Even if all you do is make copies, you'll be meeting people who might be important contacts in the future, and you'll be learning about how the game industry works. Remember to be professional at all times, even in this casual industry, and don't give up!
Resumes are important, too. If you have a good history of working on games on a volunteer or freelance basis, in addition to good internships, your chances of getting into the industry are a lot better. Once you're there, you can climb the ladder to designer, but it might take a while! Remember to market yourself and don't give up hope. This might be a high-pressure industry, but it's the right place for many people who really love games.