You can read up on how copyright applies to game development here Gamasutra - Hey, That’s MY Game! Intellectual Property Protection for Video Games
But basically copyright protection exists the moment an author fixes an expression in a tangible medium. This means the moment you save your source code to disk, or you sketch out the artwork for your game character or level art, you automatically have copyright protection without doing anything further. I would personally suggest keeping documentation of dates (the version control commits would probably work), and keeping some sort of generic readme file would help as well.
However, at that level, you may still have to put up a fight in a court case, unless your documentation is just so flawless. An author can also choose to register the copyright with the U.S. Copyright Office (current registration fee is $45), which provides certain additional benefits, such as the right to statutory damages for copyright infringement.
And as always, remember, copyright protection protects the actual artwork and sounds in the game as an audiovisual work, and the underlying source code as a literary work, but it wouldn’t prevent someone copying your idea. You’d have to look at a patent for that, but then it would have to be an extreme circumstance and couldn’t be too generic a concept or the patent office wouldn’t consider it.