Here's an updated list of the espers and their abilities. I've also included what spells are learnable, how fast they are learned, and the bonus lvl up effect for those that have it.

Each entry will be in this format:
Name
Attack name / MP cost
Damage Type
Spells ... X?? (learning amount)
Level Up Bonus
How to acquire

Alexander
Judgment / 90 MP
Holy Elemental
Pearl....x2
Shell....x10
Safe....x10
Dispel....x10
Remedy....x15
No bonus
Search Throne of Doma Castle after Cyan's dream, WoR

Bahamut
Sun Flare / 86 MP
Non-Elemental
Flare....x2
Max HP + 50%
Defeat Doom Gaze without using Vanish/Doom trick.

Bismark
Sea Song / 50 MP
Water Elemental
Fire....x20
Ice....x20
Bolt....x20
Life....x2
Strength + 2
Received as second set of 4 espers at Magi-Tek facility

Carbunkle
Ruby Power / 36 MP
Casts Reflect on party
Reflect....x5
Haste....x3
Shell....x2
Safe....x2
Warp....x2
No Bonus
Received in Magi-Tek research facility

Crusader
Purifier / 96 MP
Does 6000 to 9999 non-elemental damage to all enemies and all party members.
Meteor....x10
Merton....x1
Max MP + 50%
Defeat the 8 ancient dragons

Fenrir
Moon Song / 70 MP
Casts Blink on all party members (next two attacks miss)
Warp....x10
X-Zone....x5
Stop....x3
Max MP + 30%
Defeat Phunbaba and a child will give it to you (can also be bought at auction house)

Golem
Earth Wall / 33 MP
Golem protects party from physical attacks, absorbing damage equal to the caster's max HP
Safe....x5
Stop....x5
Cure 2....x5
Stamina + 2
Buy at Auction house in Jidoor

Ifrit
Inferno / 26 MP
Fire Elemental
Fire....x10
Fire 2....x5
Drain....x1
Strength +1
Defeat he and Shiva at the Magitek Research Facility.


There you have Alexander - Ifrit. I'll post Kirin - Ramuh next time, then Rayden - Starlet, and finally Stray - Zoneseek.