Lightning and Sazh are immediately thrust into a battle with a new enemy; the Myrmidon. With new enemies, using Libra (R1/RB) won’t offer you much help. But during the battle, after testing out some attacks, you will notice it picks up any strengths and weaknesses you have exploited. After a few battles, this information should be filled and it is good practice to use Libra periodically until you know exactly how to bring an enemy down. You also get an achievement for using Libra 100 times.
Kill the above enemy in the event battle - the robot has speed on its side as it can swiftly move to and from your characters. It will also use many attacks in rapid succession, so you will need to use a Potion when your HP falls to around 70. Once you stagger it, its attacks can be interrupted more easily so that should be your priority.