A couple of things I would recommend to try would be to look for games that have a strong modding community and become an active participant. Determine what you like, IE - User Experience design, level design, networking, Front End vs Back end development, storyboarding, concept artist, etc
Go to the Careers section of your favorite developer and look at what jobs are being posted and what would sound interesting. Pick one of those careers and research what it would take to become skilled in that profession and if it sounds interesting to you. Below are a few links to look through.
I hope that helps!
My advice is to stay with it as a hobby. My brother is an executive producer at a large video game company and his wife is more in the video game app side of things. Both worked their way up from being testers, but it took nearly twenty years. My sister in law has a gaming degree, but my brother doesn't have a degree at all. The industry has changed dramatically since he first got into it, so not having a degree is unheard of.
These days the industry is pretty brutal. If you can handle the long, stressful hours or mandatory overtime, deadlines and do-it-or-else bosses for a few years, you may be able to start making a good living. But that really depends on where you live. Huge layoffs happen more often than you might imagine, and slip-ups aren't tolerated well because there's a list of eager applicants waiting to take the job.
Most people I've met over the years or know of in one way or another live in Silicone Valley or Seattle and share a place with a lot of other people. I'm talking people in their late twenties who live with five to seven roommates in a three-bedroom apartment and barely get by.
I don't want it to seem hopeless because there are definitely success stories peppered throughout the industry of people who defied the odds and ended up in a right-place-right-time situation.
I've been a college professor for twenty years and have had literally hundreds of students who wanted to work in the gaming industry. The advice my brother has consistently given me for them is that candidates with strong writing and creative and critical thinking skills, especially under pressure, make the biggest impressions.
I hope this helps.
Elisha's answer is also true. It's a tough, tough business that eats people up and spits them out. You have to really want it and be willing to endure whatever the business demands of you. That does NOT mean you have to stand for crunch time or being abused or not having a good work-life balance. There is a growing awareness of the need for QoL (Quality of Life = good work-life balance) in the industry. There are still some companies that abuse newbies who are desperate to get in, but my advice would be to avoid them and find a different way to break in. Beware of job listings where "passion for games" is used as an alternative way to say "we'll work you to death."
Now, to get more granular, your question about "game design" is vague. Game design is now a field that has splintered into many disciplines and categories. You could be a general sort of game designer or you could become a specialist. Level designers need art and architecture skills and would work mainly on AAA console games where designing the layout of levels is a significant part of the game. Systems designers need excellent math skills and are always in demand on nearly any kind of game. And so on.
Christy recommends the following next steps:
You can make a good living in any industry. You just need to be good at.
I am working with ex-game designers/visual artists now. Based on what I heard from them, it is a booming industry. But also it is a relative small and competitive industry. To give you an example, Epic Games who owns Unreal Engine, they start a new group to look for Unreal application outside game industry like medical, retail, and etc. Just something to keep in mind.