The game starts off with the main character meeting Chancellor Shepperd. From there the player meets Jaden Yuki, the best Slifer Red student, and his friends Syrus Truesdale and Chumley Huffington. The player also meets the professor of Slifer Red, Lyman Banner. In order to meet other people, the main character must trigger certain events such as entering Ra Yellow.
There are numerous ranks in the game that usually determines who the player can duel with. There are monthly exams which determine the player's rank and dormitory. There are ways the player can get exemptions from the exams.
Source: Wikipedia, "Yu-Gi-Oh! World Championship 2008", available under the CC-BY-SA License.