There are a lot of choices in this quest that will affect the outcome. Using this guide will only take you through one possible route: the one the author has taken.
We Need An Adventurer!
Start by talking to the firemakers gathered around a cave entrance south of Eagles' Peak. They will explain that Isis had a vision about a book and arcane firemaking knowledge, but they need help exploring the cave.
Agree to help and start the quest. You can choose to talk to the all of them first, to get to know them, or go straight into the dungeon. Walk forward toward the center of the room, and Flint will give you his pitch can.
The goal in this first room is to light fires to finish the shape of the arrow pointing towards the entrance. (The below screenshot is missing one of the fires needed to complete the puzzle.)
The floor will crack and you will fall in. Light the nearby logs. Look on a rock nearby for Chapter 1 of the Firemaking journal. The room will shake again, and an unknown being will make itself heard. Try talking to the unknown, and it won't be very happy, and insists on playing a game.
The fire will go out, and there will be a scream, and Phoenix will be dead. The unknown's name is Char, and she will let you through her caves, but she has taken the mind of another member of the party. In the next camp, someone else will die unless you can identify who is possessed.
Try talking to Twig, and he will suggest that Emmett is lying. He's suddenly very confident. Head to the cave to the east to find another fire puzzle. Using the symbol from the journal, complete the fire.
Rocks will begin falling from the ceiling. Avoid them, or you'll be dealt a fair amount of damage. Take the rocks and pile them up to the east.
Light the fire and take the journal. Char will talk to you and mention that she was close to Zaros. Next to the fire is a column; tie up Twig. The fire will go out, and no one will die.
In the next room, take some Yellow and Red power. Start by mixing the yellow powder, then completing the inner shape. Next mix the red powder and finish the outside shape.
A wall will then sweep across the room, with a couple gaps in it. Run to where the gaps will sweep past to avoid getting hit. The wall will then change and sweep across the room in the other direction. Head into the next room.
Like A Hot Potato...
Enter the room, then jump across the platforms to the west. You may start taking damage until you light the fire in the corner to the far west. Pull the column switch and head all the way to the east now.
Light the fire there, and pull the next column switch. Head back west, then north. Make your way through the room and hit all the switches and light all the fire pits. Finally make your way to the northeast.
The Next Choice
Enter the room, light the fire, and pick up the journal. After talking with Char, choose to tie up Sera. The lights will go out, and no one will die. Light the fire once more.
Enter the next room and complete the fire puzzle. Take the powder as needed. In the picture below, the space where I'm standing is a yellow flame.
Now enter the next room. Oil trenches will crisscross the room. The purpose of this room is simply to survive. You can light the oil trenches to block off sections, as the enemies in the room will be unable to cross the fires. Survive long enough, and you will be able to progress into the next room.
We Must Move Onward...
Enter the next room and light the fire. Grab the next journal and talk with Char once more. Finally, tie up Flint. The lights will dim, but Lina will die. (Note: You could choose to tie up Twig, but due to the author making the wrong choice, any future choices will be different than those of the author.
Head into the next room, but be warned, do not advance further into the cave just yet. Complete the puzzle. In the picture below, the space below the blue flame behind where I'm standing, and the space in front of where I'm standing should be red flames.
The lights will go out. Light the nearby fire pit and take a torch from it. Continue on down the cave as best you can. You will be dealt large amount of damage in the meantime.
Grab the next journal and talk with Char. Tie up Emmett, and no one will die. Light the fire again. Head into the next room and try to take chapter 6. You will tie everyone up onto pillars before that though. Char will finally be in her true form.
Climb down the steps when ready. Run around and light fires and attack Char. Your damage is based off how many fires you have. You can receive accuracy boosts by luring the sprites into your fires.
Char will occasionally rage. Stay away from her at this point, as you won't be able to damage her, and she will deal massive amounts of damage. If she is near the water while raging, it will cool her off. Anytime after she has half health, the walls of fire will sweep through the arena. Try to squeeze through the gaps in the wall, otherwise you will take large amount of damage.
When Char is almost dead, she will call off the fight. She will apologize and explain that she will strive to become better and serve Zaros. Climb up the steps to the north and exit through the tunnel.