One of the most common plans in the KIA is to play e5, which gains space, kicks the f6 knight and enables white to start a kingside attack. If black has played e5, you will have to remove the black e pawn before you can advance e5 yourself, making it much harder to start an attack. Other common plans involve breaks with f4 and pawn storms to disrupt the enemy king's position. c3 is a good move if your opponent has a knight in c6 because it restricts the knight from moving agressively, and also prepares b4 (starting a queenside pawn storm) in case black castles long.
The bishop in g2 is inactive for a while, but when its diagonal gets opened, it becomes one of the strongest white pieces, controlling a nice long diagonal. Also, it's an additional defender for your king, and that's important because in the KIA we usually do pawn storms with the pawns that protect our king.
I suggest starting with Nf3 instead of g3, otherwise you allow your opponent to play e5 immediately, and unless you have specific reasons to play 1. g3, 1. Nf3 will do the same thing in the end but prevent a quick e5. If you know that your opponent is a French player, you can also start with e4 because he will play e6 anyway. That said, g3 is a not bad first move, i just don't like it.