You will receive a quest called [Group] Three Keys when you reach a certain level. This quest is the quest required to be completed in order to gain access to the Fire Temple. The first thing you will need to do for this quest is speak to Anud who is located in the Morheim Ice Fortress.
After you refresh your quest at Anund you will want to talk to the Flight Transporter and fly south to Alsig Village. From Alsig Village you will want to head northwest to the Arkanis Sky Temple.
Once you get close enough to the Sky Temple you will notice you will have to fly to your quest NPC. He isn't too far away from where you have to fly.
Talk to Bulagan again to refresh your quest. Now is where you will need a group to complete the next part. Find a group of friends and head to the Water Entrance of the Sky Temple of Arkanis. The Water Entrance can be found due north of Bulagan.
Once you're inside you will watch a quick cinematic explaining to you what you need to do. Inside the Water Room you will need to kill the Water Spirit at the far end. He will drop the key to enter the Wind Room.
After you complete the Water Room and exit it back to Morheim you will want to head around the island to the Wind Room. In the Wind Room you will want to do the same thing as the Water Room, kill all the mobs and then kill the boss for the next key. This time, however after you clear all of the Earth Spirits you will need to Glide down to the room below.
Below you will find Wind Spirits. Here you will need to find the boss, his name is Fire Key Guardian. He is the only Earth Elemental down with the Wind Elementals. This boss will drop the Fire Key. Now with the Fire Key in hand you will want to go to the door closest to Bulagan. This will take you to the Fire Room.
Inside the Fire room you will need to clear the first floating island. Your best bet here is to clear around the walls. It's the safest route around. Once you get closer to the second island you will need to Glide down to it. Have everyone go as a pact because chances are you will agro something when you land.
TIP: Don't Glide for long, only use it to break your fall.
Once you're on the second island you will want to kill your way around it until you get all the way east near the 3rd island. Once you're near the 3rd island you will want to glide off and over to it. As with the other island, BE CAREFUL. Make sure everyone goes together. The boss on this platform is at the far southeastern side.
After you kill him head through the door behind him and turn in your quest to Bulagan. After you turn in your quest to Bulagan you will need to head to Alsig Village, which is the village south of the "Fiery" area on the map. Once you arrive talk to Gelkugin to complete your quest.
Gelkugin will give you a follow up quest to return to Bulagan. When you return to Bulagan you will officially be attuned for Fire Temple.
Congratulations you're now attuned to Fire Temple!