Game is signed by making both hands into a fist with your thumbs sticking up, then thumping your fists together. In ASL, the sign for challenge is similar to the sign for game, but just use a bigger motion and more body language to indicate that it is a more serious matter with challenge.
Think of players thumping fists while playing a game to remember the sign.
We would start by using game to refer to one specific game of our baby's. For example, before you play peek-a-boo, you could sign game. As your baby starts to learn the sign, you can generalize the meaning of the sign by applying it to other games you play together.
We use the sign for game to help our babies and toddlers distinguish between real-life dynamics and playing. You can start by always using the sign for game when you are to begin playing with your child.