Main Quest - A Precious Source of Power
Here is a job from Holly telling you clear the power plant of daemons with the "hunter" as your ally. The enemies are mainly level 7 Goblins and the occasional level 24 Garchimacera. You should be able to put them away pretty quickly at this point, and will be rewarded with 5 bonus AP if you do so fast enough.
Also don't forget to grab the Silver Bangle on the ground on your way out of the power plant. Holly then gives you the Mythril Component as a reward.
Main Quest - Brave New World
Before leaving Lestallum for Cape Caem, look for Iris and talk to her to receive some Moogle Plushies that you can use as decoys in battle. There are a ton of new sidequests available as well so go do them first if you like. And once you're ready to continue with the story, go talk to Cid at the lighthouse in Cape Caem then pull lever to activate the elevator. When Umbra shows up with Luna's notebook, write something back and the lift should get going.
Once the elevator reaches its destination, pick up the Heliodor Bracelet directly ahead when you exit the lift. Talk to Iris and Talcott then go downstairs. With the scene over, speak to Cor one more time and he'll give you a Magic Flask. After which you can finally head for the boat. There's a Hi-Potion at the end of the dock by the way. Speak to Cid (who is already on the boat) so you can set sail. Walk left on the dock toward the boat and a "Talk" prompt should appear. Respond to that to communicate with the man and proceed.