Yugi is a fairly average kid. He has a few friends in Joey, Tristen and Tea, and his grandfather owns a gameshop. His grandfather has taught him a lot about Duel Monsters, but his most important lesson is a trust in the deck he calls The Heart of the Cards. When one of the other students in his class learns that Yugi's grandfather has one of the ultra rare Blue-Eyes White Dragon cards, this classmate and business tycoon, Seto Kaiba, challenges Yugi's grandpa to a duel for ownership of the card. It's here that Kaiba, the current champion of the game, shows that he possesses the other three Blue-Eyes cards and with them, he tramples Gramps. Yugi then challenges Kaiba to a duel with his grandfather's deck and remarkably defeats the champion.
What most people don't realize is that Yugi has an item called the Millennium Puzzle hanging around his neck. When he finds himself in a particularly tough duel, a spirit within the puzzle comes out and joins with Yugi to help him. After defeating Kaiba, Yugi is thrust into the attention of most Duel Monster players, as well as its creator, Maximillion Pegasus. Before the dust from his battle with Kaiba settles, Pegasus challenges Yugi to a game the likes of which he has never seen before, and when Yugi fails, Pegasus steals his grandfather's soul and challenges Yugi to join his tournament. If he wins and beats Pegasus at the end of it, then he will free Yugi's grandfather.
Yugi and Joey join the tournament at Duelist Kingdom while Tristen and Tea tag along for support. Yugi is in it for his grandfather's soul, while Joey hopes to win the prize money to pay for his sister's surgery. With so much on the line for both characters, they have to show that they not only have better strategy than any of their opponents, but that their trust in the Heart of the Cards will give them just the card they need to win.
Most of the first season is the Duelist Kingdom tournament. While on the island, the pair will not only face a slew of opposing duelists, but the mysteries surrounding Yugi's Millennium Puzzle start to become known. Not only do they know that Pegasus has another Millennium item called the Millennium Eye, but it seems a schoolmate of theirs, Bakura, has something called the Millennium Ring, and when holders of Millennium Items duel, they get sucked into a place called the Shadow Realm where the monsters are real and the stakes are always higher.
Joey and Yugi end up befriending a top-ranked duelist called Mai. At first, Mai's selfish desires seem to falter in the face of the Heart of the Cards, but as she learns to think about more than just herself, she not only becomes less antagonistic, but actually helps Yugi out of a couple of binds.
Just as the tournament starts to draw to a close, Yugi faces off against Kaiba once again, only this time Kaiba isn't playing just to retain his reputation. Pegasus has stolen Kaiba's younger brother's soul as well and the only way he will let the reigning champion challenge Pegasus for the boy's soul is if he can defeat Yugi. While Yugi does get the upper hand in the duel, he becomes worried about the Puzzle's spirit's willingness to win at any cost. Kaiba forces Yugi to stand down by standing in a precarious position so that the final blow, when played out by the holographics projectors, will cause Kaiba to fall off the side of a castle. Yugi stops the spirit from acting and forfeits the match. Before he can make his way through the tournament's last few rounds, he has to regain his confidence in the Millennium Puzzle's spirit and even face his own best friend.
Yu-Gi-Oh!: The Official First Season wraps up with a few short story arcs that are more or less the result of Yugi's time at the Duelist Kingdom. There are a few episodes where we learn about some of Gramps' history, a few where Kaiba's in trouble and only Yugi's friends can help, and some where Yugi has to learn a whole new game on the fly in order to prove his performance at the Duelist Kingdom wasn't just a fluke.
Yu-Gi-Oh!: The Official First Season is the Volume 1 and Volume 2 DVDs together in a single box, so you can get the same by buying the two volumes separately, but any fan of the show will probably want to pick up the combined set at the same time, and a quick lookup at Amazon shows that buying the complete season is a little cheaper, at least at the time of this writing.