Continuity mistake: When Shifu is speaking to Oogway under the mystic peach tree, he slams the ground, causing a large number of peaches to fall. In the following wide shot, only a few peaches are on the ground - none of which are even close to Shifu, who hasn't moved very far from his original position.
Continuity mistake: When Tai Lung is at prison, and he breaks his shield-thing off, the guardians fire four arrows at him and he evades all of them. He uses the fourth and last one to break the shackle off his hand. But in the next camera angle, there are five arrows stuck in the floor. Where did the fifth one come from?
Plot hole: Zeng the duck is sent away to Chorh-Gom prison by Shifu before Shifu and Oogway discuss the dragon scroll and the dragon warrior. While the selection ceremony is underway at the Jade Palace, Commander Vachir is touring Zeng around the prison far away. Neither Commander Vachir nor Zeng knew about the decision to select a dragon warrior, but Vachir says to Tai Lung, "Hey, tough guy! Did you hear? Oogway is about to give someone the Dragon Scroll, and it's not gonna be you." While we as viewers know about the scroll (dramatic irony), none of the characters in the prison scenes knew about the dragon warrior selection, and could not have told Tai Lung. Without this revelation, Tai Lung has no motivation to escape.