Here, I will explain (roughly) how to get each sphere. It will be in an order such as:

The Sphere, The fiend that drops it, where the fiend is located and whether it's a rare drop or common drop or in some ocasions bribe.

Lv1. Sphere : Master Coerl, Omega Ruins, Common.

Lv1. Sphere : Puroburos, Omega Ruins, Rare.

Lv1. Sphere : Sand Worm, Sanubia Desert, Rare.

Lv1. Sphere: Imp, Cavern of the Stolen Fayth, Mt.Gagazet, Bribe.

Lv2. Sphere: Defender Z, Zanarkand Dome, Omega Ruins, Common.

Lv2. Sphere: Tonberry, Cavern of the Stolen Faith, Rare.

Lv2. Sphere: Varuna, Omega Ruins, Rare.

Lv2. Sphere: Behenoth, Mt. Gagazet, Bribe.

Lv3. Sphere: Behemoth King, Inside Sin, Rare.

Lv3. Sphere: Demonolith, Inside Sin, Omega Ruins, Rare.

Lv3. Sphere: Defender Z, Zanakand Dome, Omega Ruins, Rare.

Lv3. Sphere: Machea, Omega Ruins, Rare.

Lv4 Sphere : Land Worm, Inside Sin, Rare.

Lv4 Sphere : Chimera Brain, Calm Lands, Bribe.

All the next Spheres are obtained at the Monster Arena by creating monsters (Area and Species Conquest, and the Original) with the exception of the clear sphere (which is technically obtained by creating monsters, but you have to defeat Ultima Buster.)

Hp Sphere (changes an empty node into an HP+300 node), Iron Clad (Species Conquest), Monster Arena, After Battle. You can create him by capturing 5 Iron Giants (Thunder Plains), 5 Gemenies with a Star sword (Inside Sin) and 5 Gemenies with a Broad sword (Inside Sin). When Iron Clad is created you get 60 Mana Tablets.

MP Sphere (changes an empty node into an MP+40 node), Vidatu (Species Conquest), Monster Arena, After Battle. You can create him by capturing 4 Gandarewas (Mushroom Rock Road), 4 Aerogues (Thunder Plains) and with 4 Imps (Cavern of the Stolen Faith). When you create Vidatu you'll get 99 Mana Springs as a reward.

Agility sphere (changes an empty node into an agility +4 node), Fenrir (Species Conquest), Monster Arena, After Battle. Fenrir can be created by capturing: 3 Dingos (Besaid), 3 Garms (Djose Highroad), 3 Mi'ihen Fangs (Mi'ihen Highroad), 3 Snow Wolfs (Macalania Lake), 3 Sand Wolfs (Sanubia Desert), 3 Skolls (Calm Lands) and 3 Bandersnatchs (Mt. Gagazet). Then you'll be rewarded with 99 Chocobo Feathers.

Evasion sphere (changes an empty node into an evasion +4 node), Pteryx (Species Conquest), Monster Arena, After Battle. Pteryx is created by capturing 4 Condors (Besaid), 4 Simurghs (Djose Highroad) and 4 Alcyclones (Sanubia Desert). When Pteryx is created you'll get 99 Mega Phoenixes.

Accuracy sphere (changes an empty node into an accuracy +4 node), Hornet (Species Conquest), Monster Arena, After Battle. Hornet is created once 4 Killer Bees (Kilaka Forest), 4 Bite Bugs (Djose Highroad), Wasps (Macalania) and 4 Nebiros (Calm Lands) have been captured. Then you'll be rewarded with 60 Mana Tonics.

Magic Def. Sphere (changes an empty node into a Magic Def +4 node), One-Eye (Species Conquest), Monster Arena, After Battle. You can create One-Eye by capturing 4 Floating Eyes (Mi'ihen Highroad), 4 Buers (Thunder Plains), 4 Evil Eyes (Macalania Lake), 4 Ahrimans (Mt. Gagazet) and 4 Floating Deaths (Omega Ruins). When created you will be rewarded with 60 Stamina Tablets.

Magic Sphere (changes an empty node into a Magic +4 node), Jumbo Flan (Species Conquest), Monster Arena, After Battle. Jumbo Flan (Species Conquest) is created by capturing 3 Water Flans (Besaid), 3 Thunder Flans (Mushroom Rock Road), 3 Snow Flans (Djose Highroad), 3 Ice Flans (Macalania Lake) and 3 Dark Flans (Mt. Gagazet). When Jumbo Flan is created you will get 60 Twin Stars.

Defence Sphere (changes an empty node into a Defence +4 node), Tanket (Species Conquest), Monster Arena, After Battle. Tanket is created by capturing 3 Raldos (Mi'ihen Highroad), 3 Bunyips (Djose Highroad), 3 Murussus (Macalania Woods), 3 Mafdets (Macalania Lake), 3 Shreds (Calm Lands) and 3 Halmas (Omega Ruins). When created you'll get 99 Golden Hourglasses in return.

Strength Sphere (changes an empty node into a strength +4 node), Juggernaut (Species Conquest), Monster Arena, After Battle. Juggernaut is created by capturing 5 Dual Horns (Mi'ihen Highroad), 5 Valahas (Cavern of the Stolen Fayth) and 5 Grendels (Mt. Gagazet), when you create Juggernaut you'll get 99 Light Curtains in return.

Teleport Sphere (when you use a Teleport Sphere you can go to any node already activated by another character), Sleep Sprout (Species Conquest), Monster Arena, After Battle. Sleep Sprout is created by capturing 5 Furguars (Mushroom Rock Road), 5 Thorns (Cavern of the Stolen Fayth) and 5 Exorays (Inside Sin). When created you will be rewarded with 99 Healing Springs.

Return Sphere (when used you can travel back to any node you have already activated with that character), Chimerageist (Area Conquest), Monster Arena, After Battle. Chimerageist is created by capturing all the monsters in the Calm Lands (Nebiros, Skoll, Flame Flan, Shred, Anacondour, Coerl, Ogre, Chimera Brain and Malboro). When created you will get 60 Farplane Winds.

Friend Sphere (move to any node that any character is on right at that moment), Vorban (Area Conquest), Monster Arena, After Battle. Vorban is created by capturing all the fiends in the Omega Ruins (Floating Death, Black Element, Zaurus, Spirit, Halma, Master Coerl, Machea, Puroburos, Wraith, Both Gemenis, Defender Z, Demonolith, Master Tonberry, Adamantoise, Varuna and Great Malboro). When Vorban is created you will be rewarded with 60 Designer Wallets.

Fortune Sphere (activates a Luck node), Earth Eater (Original), Monster Arena, After Battle. The Earth Eater can be created once two area conquest monsters creations have been created. When Earth Eater is created you'll be rewarded with 60 Three Stars.

Luck sphere (changes an empty node into a Luck +4 node), Greater Sphere (Original), Monster Arena, After Battle. Greater Sphere can be created when two monster creations in the species conquest have been created. When its created you'll get 60 Supreme Gems.

Warp Sphere (lets you move to any sphere in the whole grid), Nemesis (Original), Monster Arena, After Battle. Nemesis is created once you have captured ten of every monster in the whole game and beaten all the Area and Species Conquest and the Original creations. You can also get 99 Warp Spheres by solving the 12 chests in Omega Ruins. When you create Nemesis you'll get 10 Master Spheres (which when used – you can activate not move any node in the whole grid).

Clear Sphere, Erases an attribute node such as Strength, Defence, HP, Magic, Magic Defence, MP, Agility, Evasion, Accuracy and Luck. But it won’t erase Skills, Specials, and Magic. To get Clear Spheres beat Ultima Buster. (Beat him once, then buy Clear Spheres from Arena owner for 10,000 gil). Ultima Buster can be created when 5 monsters of each area of the whole game are captured. When you create Ultima Buster you will get 99 Dark Matters in return.

Ability, Power, Mana and Speed Spheres can be obtained by feeding Kottos Distillers (Power, Speed, Mana and Ability). Kottos is a monster creation which can be created by capturing all the monsters on Mi’ihen Highroad (Floating Eye, Mi’ihen Fang, Ipiria, Raldo, Vouivre, White Element, Thunder Flan, Bomb and Dual Horn. When Kottos is created you will be rewarded with 99 Soul Springs.

A Skill Sphere will let you activate a Skill node already activated by another character. These can be obtained by bribing a Zu in Sanubia Desert.

A Special Sphere will let you activate a Special node already activated by another character. You can get these by bribing an Adamantoise which are found in the Omega Ruins.

An Attribute Sphere will let you activate an Attribute node already activated by another character. These can be obtained when you bribe a Maelspike in the water areas of Mt.Gagazet.

A Wht. Magic Sphere will let you activate a Wht, Magic node already activated by another character. These can be obtained by bribing a Dark Flan in Mt.Gagazet.

A Blk. Magic Sphere will let you activate a Blk, Magic node already activated by another character. You can get these by bribing a Black Element in the Omega Ruins.

And finally the Master Sphere which will allow you to activate any node regardless of location. You can obtain the Master Sphere by capturing 10 of each of the fiends in the whole game. Then bring them to the Monster Arena and you will be rewarded with 10 Master Spheres. Or you can get Master Spheres with a Rare drop from a Dark Aeon (2 for an overkill). And Penance’s arms will drop them as a Common Drop each(2 for overkill).

Hope this Helps =)

'The Source