Let this serve as a guide to complement Nyxil's solution, which this it based on. The spreadsheet is great to get a sense of what you're missing without spoiling locations, but for those who want more detailed instructions on how to acquire anything, this is for you.
If anything is wrong or any vital information is missing, please feel free to comment and I will edit the solution. Some locations are missing at the moment, so if you have anything to contribute, please do.
Merchants you will need to find for this achievement:
Rosabeth of Melfia - Petrified at the entrance to the Shaded Woods, reached through a cave in Majula. Use a Fragrant Branch of Yore to bring her back to life (these are found throughout the game). You'll have to go here eventually, so she is technically not missable. Be careful of pulling her lever before she has relocated to Majula, as the enemy horde it summons may end up getting her killed.
Straid of Olaphis - Petrified in a cell in Lost Bastille, up the stairs from the room on the Bastille side of the bridge to Sinner's Rise. He never moves from this location.
Titchy Gren - The leader of the Brotherhood of Blood covenant. To find him, you'll have to cross the long bridge from Huntsman's Copse to the Undead Purgatory. Defeat the boss here (Executioner's Chariot), then you can continue up the stairs to find him. You will need to join the covenant to access his wares.
Chancellor Wellager - Right inside the gates of Castle Drangleic, a mandatory area. Just go up the stairs and he's right there, in ghost form. He never moves from there, but you will need to go to New Game++ to access some of his wares.
Grave Warden Agdayne - In Undead Crypt, shortly after the room with the mages after the first bonfire. When you get near his room, he will warn you to put out any lights you have with you. Do this, or he will turn hostile. As long as you remain in the dark, he will talk and sell things to you.
Now for the actual spells:
Fireball - Bought from Rosabeth (1200 souls).
Dropped by Salamanders in Forest of Fallen Giants. These guys are behind the door just before where you fight the Last Giant. You will need the Iron Key from the first room of the Iron Keep to unlock the door.
Found in No-Man's Wharf. In the chest right after defeating the Flexile Sentry aboard the ghost ship. Just climb the ladder and turn around.
Fire Orb - Bought from Rosabeth (3400 souls).
Dropped by Salamanders in Forest of Fallen Giants. See Fireball for location.
Great Fireball - Dropped by Salamanders in Forest of Fallen Giants. See Fireball for location.
Found in Brightstone Cove Tseldora. After killing the boss of this area, keep going and you'll find a little office of sorts. Kill the enemy here and he will drop a key. Use this to unlock the door back in the big room with spiders hanging from the ceiling, just before the even bigger room with all the spiderwebs. In here, after you kill a bunch of spiders, you'll find a chest with, among other things, this pyromancy.
Great Chaos Fireball - Reward for reaching Rank 3 in Brotherhood of Blood Covenant. This is not recommended as it requires huge amounts of PvP wins.
Bought from Chancellor Wellager in NG++ or later (17400 souls).
Firestorm - Bought from Titchy Gren (4500 souls).
Dropped by Poisonous Horn Beetle in The Gutter. Near the end of The Gutter there is a ladder you can climb up into a cave containing this giant beetle along with several of the smaller poison creatures.
Found in Drangleic Castle. In the really big room with archers on balconies and stairs leading up to them, this is in a chest on one of the balconies/upper floors.
Fire Tempest - Dropped by Guardian Dragon in Dragon Aerie. There are only three of these dragons and they do not respawn, so if you want more, either go into NG+ or use a bonfire ascetic on the area to respawn them.
Found in Shrine of Amana. (ADD LOCATION)
Chaos Storm - Found in Iron Keep. First off, make sure you have the Flash Sweat pyromancy. Attune it, and put on your best fire resistant gear. Now make your way to the big room near the end of the Iron Keep, where you can raise and lower a big central platform into lava. Make sure all the enemies are dead, then lower the platform. Climb to the upper level of the room and you'll see some urns there. These contain water which you'll need to douse yourself in to get some additional fire resistance. Smash them from right to left until you're soaked, then drop down to the middle level, then down to the lower level, on the left side of the room. Go down the stairs here, cast Flash Sweat on yourself, and run over to the chest at the end of the lava rocks. You should at this point be able to resist enough damage to safely retrieve the spell within.
Combustion - Bought from Rosabeth (1500 souls).
Found in Huntsman's Copse. If you go right from the entrance to the Skeleton Lords boss fight and enter the cave, you can find this past a tunnel after the necromancer and skeletons. You can see it from outside, it's the corpse hanging out of a cave opening.
Great Combustion - Bought from Titchy Gren (4800 souls).
Found in Drangleic Castle. Just outside the main gates of the castle, over on the right side in a chest.
Fire Whip - Bought from Titchy Gren (3800 souls).
Dropped by Bell Keeper Phantom in Belfry Sol. Belfry Sol is reached by using a Pharros' Lockstone on the contraption in small room found through a door in the same room of the Iron Keep where Chaos Storm is found. The little black dwarf phantoms here drop pyromancies, although very rarely.
Poison Mist - Bought from Rosabeth (3400 souls).
Toxic Mist - Acquired by trading the soul of the Royal Rat Vanguard with Straid (1500 soul cost). The Vanguard is encountered in the Grave of Saints, reached through the pit in Majula.
Acid Surge - Acquired by trading the soul of the Royal Rat Authority with Straid (1500 soul cost). The Authority is encountered in the Doors of Pharros area which you'll pass through on your way to Brightstone Cove Tseldora, and is fittingly enough best fought using pyromancy.
Lingering Flame - Bought from Straid (6700 souls).
Dropped by Undead Jailer in Lost Bastille. These are the big cloaked guys with halberds/axes.
Flame Swathe - Bought from Straid (9500 souls).
Dropped by Bell Keeper Phantom in Belfry Sol. See Fire Whip for location.
Forbidden Sun - You will have to complete a questline for Royal Sorcerer Navlaan in Aldia's Keep for this. You must be hollow to talk to him, and he is found by entering the keep, going to the end of the ground floor and following the hallway right. Going left leads to a lever that releases Navlaan from his cell, so don't pull it. He will give you three tasks before giving you the spell. First, he wants Gilligan's ladder miniature. At this point in the game, Gilligan is most likely sitting by the pit in Majula, where you can just talk to him and buy the miniature (ADD PRICE). Next, he wants the helm of Cale the cartographer. If you encountered him in Forest of Fallen Giants, he can now be found downstairs in the manor in Majula. To get his helm, all the flames on the his map need to be lit. To achieve this, just light most of the bonfires and kill all the main bosses. If all flames are lit (which also gives you the Curious About Maps achievement), talk to Cale again and he will give you his gear, incuding the helm. Finally, Navlaan wants the staff of Felkin, the Sunset Staff. Felkin, a merchant right before the first bonfire in Huntsman's Copse, will give you the staff if you talk to him with at least 20 intelligence AND 20 faith. Otherwise, you can kill him for the staff (he is required for some hexes though), or find one in Aldia's Keep by killing the mimic outside. After all this, he will give you Forbidden Sun. Also not that you actually do not lose the items, you merely show them to Navlaan.
Flame Weapon - Acquired by trading the Old Witch Soul with Straid (10,000 soul cost). This soul is dropped by the Lost Sinner only in NG+ and onward.
Flash Sweat - Bought from Rosabeth (2300 souls).
Iron Flesh - Bought from Rosabeth (3500 souls).
Warmth - Bought from Agdayne (5200 souls).
Dropped by the rats in Grave of Saints.
Immolation - Found in Belfry Sol. See Fire Whip for how to get there. First, you have to ring the bell in the back left tower to open a gate further on. Climb the ladder in the other tower, go across the rooftop, drop down at the end, and go through the doorway there to find the chest with the pyromancy.