The game chromatic number χg is investigated for Cartesian product G□H and corona product G∘H of two graphs G and H. The exact values for the game chromatic number of Cartesian product graph of S3□Sn is found, where Sn is a star graph of order n+1. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let Pm be the path graph on m vertices and Cn be the cycle graph on n vertices. We have determined the exact values for the game chromatic number of corona product graphs Pm∘K1 and Pm∘Cn