Welcome! Game design is an endlessly fascinating and exciting field. It’s awesome that you want to try it out, and you can get involved without needing to invest anything more than your time and curiosity.
A lot of people will tell you start by learning something like Unity or GameMaker. Those are both good programs, but there’s a certain amount of lead time involved as you learn them. My suggestion would be to start by creating some games; once you’ve gotten a sense for where you want to go with your first design(s), you’ll be better positioned to judge which tools will help you get there.
Instead of working on the computer, begin by learning with Magic: the Gathering. Try making up some cards, and then play them and see how they work out. (Don’t worry about making them pretty–just cut up slips of paper and sleeve them with normal cards.) Expand that into a full set, or design a cube for drafting. Get used to reaching into a game’s systems.
While you’re doing that, play some games that will give you a sense for just how vast the genre is, and how limitless the opportunities for creativity are. Gone Home, Proteus, Beyond Eyes, Papers, Please–the list goes on. Challenge yourself to see games in new ways.
Don’t neglect to read, either. There’s a lot of great writing about game design out there. The links page above has some; when you’ve exhausted that, I’d recommend Salen and Zimmerman’s Rules of Play.
Game design is sometimes fun, always challenging, and incredibly rewarding. I hope you enjoy it as much as I do, and that the advice above helps you get started. Again, welcome!