Story mode starts out with your avatar being introduced to the game of Go by your personal Go master. After teaching you some basics, he sets you on the Path of Go where you will face off against other masters and look for a long lost relative (it's a slight spoiler, so I'll say relative rather than the actual relation).
The other masters, and your relative will have you solve a couple Go problems to teach you how to handle certain situations, and beat them in games, sometimes as black, sometimes as white, and sometimes with a handicap. Early on they even let you undo your last move, but that ends quickly.
All in all you will play 18 matches of Go (all on 9x9 boards) and solve 8 puzzles (plus the tutorial). The difficulty gets harder as you go, but with patience (and if you have to, a computer program to plug the moves into) the masters can be beaten.
Also to note, there are 5 different areas on the Path of Go, and with each comes a different table/stone set and background for you to use in local or online play.