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.
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