As you enter Luca, you’ll learn that Blitzball season has finally begun, allowing you to build teams and play Blitzball, much like in the original FFX (although you may find FFX-2’s to be worse, mostly due to the inability to control your players). Whatever the case, you may as well play the game and hope you win the AP Egg . But you’re not really obligated to do it unless you want the Trophy attributed to it.
Head towards the Outskirts of town and get the three Light Curtains from the chest there. Approach the Save Sphere at the bottom of the stairs and go south from there to find a Moogle. You’ll soon start up a mission wherein you have to capture said "furtive, floating, fluffy, flying" Moogle. Not to mention cute.
When you regain control, go down the stairs further to the Square. Approach the Moogle and press X to start up one scene, then go south another and repeat the process. After the Moogle leaves, head south an area and southwest another to Dock 2 to find him again near the ship. Approach him and press X to start up another scene and to end the mission.