Mission 39: Collect hearts
Eliminate Heartless in Olympus Coliseum and collect hearts.
Roxas goes to collect hearts at Olympus Coliseum. He finds out from Phil that there is a giant Heartless in The Coliseum, and Hercules is nowhere to be found. As Roxas fights, he continues to have flashbacks of a boy in red. After he defeats the Heartless, Phil coaxes Roxas into training. When Roxas leaves, Hercules comes back and finds out from Phil that Roxas took care of the Heartless problem.
You'll start out outside, and will quickly be bombarded by two Flare Notes and a Jumbo Cannon. After defeating them, Phil comes for you, telling you there's a bigger Heartless to take out. The Heartless is a Clay Armor, which tend to be weak to Thunder Magic, if you've come well-equipped.
Once the Armor has been taken out, Phil says it's time for training. Gather yourself up, snag the goodies in the Vestibule (including a surprisingly easy to miss Unity Badge near the western ceiling), and then tell Phil you're ready to train.
The training this time is, like the previous one in the Coliseum, smacking barrels into Heartless, in this case, lots of Soldiers. As before, hit as many as hard and as fast as you can, and pay attention for the darker colored but very useful exploding barrels to take several out at once. Outright destroying the Soldiers with your Keyblade will not help your score, and watch your aim, as the crescent nature of this course lends itself well to accidentally smacking barrels into other barrels, doing you little good and resetting any chains you might get from eliminating multiple Soldiers in a row. You will have one minute and twenty seconds to hit as many barrels into the Soldiers as you can, and you can play this game a total of three times per mission. In order to reach the optional goal, you simply have to destroy as many Soldiers as possible during this portion of the mission; it's not a terribly high amount of Heartless and you should be able to reach it easily within three rounds. Scoring at least 130 points on this mission will reward you with a Blazing Gem when talking to Xigbar on Day 171.
Once you have enough points (or have run out of rounds), it's time to RTC.
TIP: If you're looking to snag a lot of easy Heart Points for the more expensive items this mission is very helpful to that cause. Bring lots of Aero and Ethers/Elixirs to restore them. Wait for the Soldiers to get close and blast them with Aero. Once you get into a rhythm, it should be easy to nab Heart Points from this mission. As you have three rounds to do this in, you can rack up anywhere from 1000-2000 Heart Points per mission, allowing you to nab the Moogle's items faster.
The hardest part of fulfilling this condition comes with the fight with the Clay Armor; lots of Thunder magic and locking-on should help. As for the Soldiers, you don't technically need to defeat any of them to meet the mission condition for all three badges, so don't waste your time and effort trying to hit barrels at them. Just hit them with your Keyblade if they get too close and you should be fine.