Dark Souls

*This solution contains spoilers.

Finding all the pyromancy spells can be difficult because you are required to not only acquire the Pyromancy Catalyst if you are not a Pyromancer, but also dole out all the souls required to upgrade it to +10. This is needed because a certain Pyromancer NPC, Quelana is Izalith, will not show up until you do that.

First off, I'll list all the NPC's required for this and how to interact with them.

-Laurentius of the Great Swamp: Laurentius sells many of the "basic" pyromancy spells. The thing is, this NPC is completely missable. You could even kill him by accident if you are careless. You'll find him in a room in the Depths just after you kill the first Butcher. He is trapped in a room full of barrels. He is actually stuck in one of these barrels in a corner. Do not hit the barrels with your weapons to free him. This risks you hitting him and turning him hostile forever. Instead, roll into the barrels to break him loose. He'll then head to Firelink Shrine. Go on to defeat Quelaag in Blighttown Swamp. Immediately after this, head to Firelink Shrine and talk to Laurentius. Level up your Catalyst to +10 and buy any spells you do not own. After a certain amount of time, or if you talk to him after encountering Quelana, Laurentius will go to Blighttown and turn Hollow, barring you from interacting with him anymore. Aside from you stabbing/slicing/piercing/spelling him to death.
Spells bought from Laurentius:
-Fireball
-Fire Orb
-Combustion
-Iron Flesh
-Flash Sweat

-Eingyi: Eingyi is located in Quelaag's Domain behind a secret wall. He is an egg carrier, but he is not hostile. Answer "Yes" to the question he asks and you will be let into the chamber that has a covenant leader as well, Daughter of Chaos. Eingyi will not sell you any spells unless you have been infected with the egg status. To get this, attack an Egg-Carrier enemy in Qualaag's Domain to make it grab you. When this happens, you'll be infected and eventually, eggs will grow on you. Eingyi will now sell you his spells. As well as egg vermifuge, the item that gets rid of the disgusting beast now growing on you. He also ascends your Pyromancy Catalyst if somehow you saved Laurentius, but you can no longer use him.
Spells bought from Eingyi:
-Poison Mist
-Toxic Mist

-Daughter of Chaos: She is located in the same room as Eingyi. You need to join her covenant if you want to get all the spells. Upon joining, you receive Great Chaos Fireball. To get her other spell, you need level up her covenant by offering humanity. Upon reaching level 2, you will receive the spell, Chaos Storm.

-Quelana of Izalith: Quelana will show herself in Blighttown Swamp once you have done several things. You must have saved Laurentius, defeated Quelaag, rang the second bell, and you must have leveled your Pyromancy Hand to +10. Once you have done this, Quelana can be find in Blighttown Swamp, sitting next to one of the large stone pillars near the entrance to Quelaag's Domain. She will sell you several spells and even level your Pyromancy Hand to +15, then ascend it to an even stronger version. Once you have placed the Lordvessel on the Firelink Altar, return to Quelana. She will give you a special mission to kill her mother, Bed of Chaos, the boss of Lost Izalith. Once you've completed this, she'll give you the incredible Fire Tempest spell.
Spells bought from Quelana of Izalith:
-Fire Orb
-Great Fireball
-Firestorm
-Fire Whip
-Combustion
-Great Combustion
-Undead Rapport
-Fireball

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These spells must be found.

-Fire Surge
Dropped by Engorged Zombie in the Painted World of Ariamis.

-Acid Surge
Found on a corpse in Painted World of Ariamis.

-Power Within
Found on a corpse in Blighttown.

-Chaos Fire Whip
Found in a chest in Lost Izalith.

Good luck! Any and all comments/questions are welcome!

Ultimate video guide for Bond of a Pyromancer achievement, it will show every spell location and action how to get it. Also, I don't take any credit, I just want to help you guys get it as easy as possible.

Some important characters:

Laurentius (Pyromancy trainer) - Once you beat the Capra Demon, you'll obtain the Key to the Depths, once you reach a large room with a bunch of tables (above a butcher), you'll be able to jump off the sides, do so and continue around until a butcher jumps down to attack you. Kill him and continue into the room he was blocking. You'll here him asking you to help him out of a barrel. You can roll into it or use your left hand to punch it and break it He'll thanks you by giving you the Pyromancy Flame at the Firelink Shrine. If you show him either ANY Chaos spell or a +15 Pyromancy Flame, he'll ask where you got it and leave.

Quelana (Pyromancy trainer) - There are some speculations on how to get her to appear, the most widely believed methods are to upgrade your Pyromancy Flame to +10, OR defeat Quelaag and discovering the Chaos Servants. Once you have done one or both of these, she will be near a giant pillar near the waterwheel leading to the Valley of the Drakes. If coming from the VoD and facing Quelaag Lair, there should be a one pillar in-between you and the Lair, she's near that.

Finding the Chaos Servants - After defeating Quelaag and ringing the bell, there will be some stairs that will bring underneath where you just were. You'll notice a wall that is covered in eggs and has a small egg free spot, attack it and it will disappear. There will be an egg infected NPC (Eingyi) in front of you, DON'T ATTACK HIM. He'll ask if you're here to serve, say yes and he'll move aside allowing you to join if you want.

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Fireball - Default for Pyros, purchasable from both trainers.

Fire Orb - Purchasable from both trainers.

Great Fireball - Purchasable from Quelana.

Flash Sweat - Purchasable from Laurentius.

Combustion - Purchasable from both trainers.

Great Combustion - Purchasable from Quelana.

Fire Whip - Purchasable from Quelana.

Chaos Fire Whip - Found in Lost Izalith, before the last Fog Gate (after you get attacked by a Pyro, keep going, there's a giant octo thing guarding it)

Fire Surge - Found in the Painted World of Arimias, once you find the Annex Key in the well, continue navigating the small tunnels until you find some stairs going into a small courtyard. In the corner there will be a Bloated Hollow crying in the corner. Kill it and it'll drop this.

Acid Surge - Found in the Painted World of Arimias, once you find the cemetery (where Jeremiah will invade if you're human), go to the right (if looking into the cemetery) and hug the cliffs, you'll be able to go behind a small wall where this is located on a corpse.

Fire Storm - Purchasable from Quelana.

Chaos Storm - Reach Rank 2 in the Chaos Servant covenant.

Fire Tempest - Once you meet Quelana, continue talking to her until she asks you to kill her sisters (NOT including the White Spider, Chaos Servant leader), once you kill them, she'll give this to you (you may have to buy all of her spells for her to ask this of you). She'll also drop it if you kill her, so if you've bought all her spells, you can kill her.

Great Chaos Fireball - Obtained for joining the Chaos Servants covenant.

Power Within - In Blighttown, under the giant wall parasite. If you're entering from VoD, you'll see it across the area, but you'll have to work your way up to get to it. If coming from the Depths, it's a bit after the Bonfire on the underside of a giant bridge (looked like a bridge anyways, stone and grey-ish).

Iron Flesh - Purchasable from Laurentius.

Poison Mist - Found in the Swamp, a bit before the hordes of leech like enemies, and purchasable from Eingyi, you must be infected with "egghead" to buy from him (to get infected, go to the aggro egg enemies in the Demon Ruins, let them attack you repeatedly, if you notice your character scratching the back of his head/neck, you're infected, wait a while (5-10 minutes) and it'll hatch).

Toxic Mist - Purchasable from Eingyi.

Undead Rapport - Purchasable from Quelana.


If there are any questions, let me know. If there's a request, I can make some short videos which may explain some of these things better, just let me know which are giving you problems.

Basic information taken from: http://darksoulswiki.wikispaces.com/Pyromancies