All the best game designers do it practically, by iterating on prototypes. I extended this concept to evolve designs over several games. See for example this devlog for one of mine, and the older concept it grew from. Another way is to take a game you like in principle but not in practice, and try to fix its design issues.
As for how to organize your thoughts, try different tools, like an outliner (I made a couple), or a wiki, and see what works for you. But don't get hung up on them. A plain text file goes a long way. It's more important to practice expressing yourself.
Speaking of which: don't "practice". Don't make throw-away stuff on purpose, just for the exercise. Work on the games you want to make, the art you want to make. If it doesn't come out so great, oh well, you'll do better next time. That is practice. Otherwise how are you going to care?