While investigating Olympus Coliseum for the first time, Xigbar explains the concept of the Nobodies to Roxas. Roxas also has more flashbacks of Sora. Meanwhile, Xion and Axel discuss Xion's strange feelings regarding the beach and sea.
Welcome to the newest world... the hero-crazy Olympus Coliseum. You'll arrive outside the Coliseum with Xigbar, but before you can explore, you have to fight off a few Heartless... Flare Notes and Li'l Cannons to be specific. Once you finish them off, you'll be able to explore after a cutscene. Xigbar leaves you at this point, so you will be going solo from here on out. Areas you should investigate before heading inside include the two statues, the leader boards on the south walls, the door to the south, and the east wall, towards the southern end. Once you've checked those out, head inside.
Going inside, check out all four trophies, and then answer the questions with "some kind of arena" and "Hercules?". Afterward, you'll achieve your first breakthrough. Then, go talk to Phil, who will set you training. After a quick tutorial, you'll play a barrel smash game for points. The game is a minute long, and the more barrels you smash and the faster you smash them will earn you points. Just like when you have Heartless, a heartchain ring counter will appear around the barrel you hit, and the larger it is when you hit, the more points you get. The darker brown barrels will explode, taking several out at once. The barrels home in on each other pretty well, so don't worry about hitting them in the exact direction of other barrels. You can play the game as many times as you want.
Once you decide to stop, you're free to RTC, but you still have an optional objective that requires further exploration, so you might as well go for it. In the Vestibule, check out the two gold trophies (one on each wall), then go outside and look at the giant statue near the east wall, and the Hercules cup board in the southwest corner. And there you go, optional objective complete; RTC to finish the mission.