Solve the Destruction Sphere-related puzzle in each of the main five Cloisters of Trials. Then go to the Baaj Temple.
Blitzball Easy Win Bug
Whenever you are playing Blitzball, and currently are winning (i.e. your score is higher than the other team's) just swim behind your own goalie. Stay there and everyone else on the field will go to the other side and swim in circles. Sometimes, a member of the opposite team will follow you. Just shake him loose (breaking through him is an easy way) and he too will just leave you alone. Stay there for the remainder of the game and you will win.
Secret Farplane Scene
After you get the Airship go back to Guadosalam. Follow Tromell up to the Farplane. Talk with Lulu to view the secret scene.
Skip Djose Temple
When you initially arrive at Djose Temple, you'll find Lucil riding a chocobo. You need to get in her way so as she approaches the temple; if done correctly, Tidus will glide to the area where a cutscene occurs. Skip Lucil's text box and you should be able to freely move during the cutscene. If you move toward the other road next to the one leading to the temple, you can skip Djose Temple: however, note that this will prevent you from getting Ixion and the treasures within the temple.
The Magus Sisters
Obtain the Blossom Crown and Flower Scepter and then open the door at the back of Remium Temple.
Yojimbo
Enter the Cavern of the Stolen Fayth in the Calm Lands after getting the Airship.
Anima...