Please refer to Author's Notes for clarification and answers to some important questions that may be asked.

The simple information starts at the "Very Easy" section and onwards, Starter section is detailed for the new players having trouble deciding which Starting Eidolon to choose. This guide was put together in order of easiest to obtain to hardest to obtain. This also includes the methods of obtaining each Eidolon's fragments - some of which may reflect my opinion rather than statistical truth; however those statistics are very circumstantial, just like luck.

For example how a Guild can use the Guild Hall to summon Eidolons³, which cost 100 Eidolon energy each summoning device (yet some guild's even spawn 50+ at once on some day of the week), and chooses from a random pool of the Eidolons that can normally spawn there. While other guild's never do Eidolon Summoning.

I once got lucky enough to get 3 Uzurial key fragments from a single Guild Hall Spawn.

Another example of ease would be like how Beating Zaahir roughly 40 times can only yield 3 key fragments (a personal experience), however if you were really aiming for it, its effort equates to at least 4 Infernal Abyss runs over a span of 32 hours, but 8 hours between each dungeon reset (allowing 1 entry per reset), per 20 minute run (time allowed to keep progressing through the rooms and floors). Endora would have a chance to spawn, and a chance to get her key fragment, on top of doing between 3 of the otherworld dungeons (or 5 dungeons if you include dragon dungeons) - which would have to be repeated every 2 hours. Vermillion is even harder than Endora since Hell mode is harder than party mode. All while depending on how lucky your feeling between the Eidolon spawns and actually getting the fragments.

Serif, Merrilee, Grimm, and Alessa are all "selected" upon character creation. They cannot be shared between characters, you cannot obtain several starters on the same character, and some are more desirable than others. Keep in mind this is my opinion (which is based on how good they are in a more general perspective; or their adaptability), and does not account for their skills.
Also note that starter Eidolons are the only Eidolons that do not give fragments of their keys, but instead give Hero emblems. This is because there is no use in having a key for a starter Eidolon.

1. Serif
   
2. Grimm
   
3. Merrilee
   
4. Alessa

Needlessly detailed information on why the list as is:
Please keep in mind these star buff stats are derived from before 4 star is/was implemented

Serif is most diesirable for -50% crit damage to you, and up to +30% max HP with the star buffs.

Grimm in second for +15% dmg against flame targets and up to +24% DMG.

Merrilee and Alessa tied for a metaphorical last. Alessa gives +25% normal attack speed and up to +15% chance to double attack. While Merrilee gives +5% move speed +5% give and recieved healing, and +2% of dmg dealt by skills to HP (80% less effective against certain bosses).

Sigrun:
Character quest at level 25 (full key)
Connect to Gaia (fragments only)
Guild Hall Spawns
Temple of Eidolons

Gigas:
Character quest at level 40 (full key)
Connect to Gaia (fragments only)
Guild Hall spawns
Temple of Eidolons

Eligos:
Guild Hall Spawn
Soul Box Fusion
Temple of Eidolons

Bel-Chandra:
Guild Hall Spawn
Soul Box Fusion
Temple of Eidolons

Yarnaros:
Guild Hall Spawn
Soul Box Fusion
Temple of Eidolons

Bahadur:
Guild Hall Spawn
Soul Box Fusion
Temple of Eidolons

Tigerius Caeser:
Guild Hall Spawn
Soul Box Fusion
Temple of Eidolons

Aelius:
Guild Hall Spawn
Temple of Eidolons

Uzurial:
Guild Hall Spawn
Temple of Eidolons

Vayu:
Guild Hall Spawn
Temple of Eidolons

QuelKulan:
Guild Hall Spawn
Temple of Eidolons

Kotonoha:
Guild Hall Spawn
Temple of Eidolons

Zaahir:
Infernal Abyss
Paragon Table

Nazrudin:
Whirlpool Abyss
Paragon Table

Abraxis:
Archaeology hidden treasure chest (Mysterous Treasure map)

Cyril:
Guild Instance exclusive

Endora:
Otherworld: Alabastern Temple (party mode)
Otherworld: Gydale Glen (party mode)
Otherworld: Miner's Steppe (party mode)
Lament of Thunder Dragon King
Landing of the Sky Dragon King

Vermillion:
Otherworld: Alabastern Temple (hell mode)
Otherworld: Gydale Glen (hell mode)
Otherworld: Miner's Steppe (hell mode)
Lament of Thunder Dragon King
Landing of the Sky Dragon King

Serena:
Wrathborne Temple (hell mode)
Ozymand Temple (hell mode)
Lament of the Thudner Dragon King
Landing of hte Sky Dragon King
Paragon Table

Tsubaki:
Deep's Cavern (hell mode)
Candeo Core (hell mode)
Ghostweep Cave (hell mode)
Lament of the Thunder Dragon King
Landing of the Sky Dragon King
Siege of the Aqua Dragon Queen

Hel:
Sky Tower Floor 18
Special Limited time Promotions ( ex. Eidolon Blessing Event)

Bealdor:
Sky Tower Floor 20

Kaiser Zeta:
Titan's Root (hell mode)
Cadilla Felwood (hell mode)
Paragon Table

Shirayuki:
Titan's Root (hell mode)
Cadilla Felwood (hell mode)
Sanctuary of Eidolons (needs confirmation)
Special Limited time Promotions

Cesela:
Special limited time promotions
Paragon Table
Temple of Eidolons (special occurance)

Alucard:
Sanctuary of Eidolons

Astraea:
Sanctuary of Eidolons

Kusanagi:
Special limited time promotions
Paragon Table

Diao Chan:
Special limited time promotions
Paragon Table

Hansel & Gretal:
Speical limited time promotions
Paragon table

Cleopawtra:
Founders edition purchase exclusive.

Each owned Eidolon can give you more of their own key fragments, special exception to starter Eidolons, this is because you never need their keys or key fragments (and you are unable to have several of the starter Eidolons). For example if you have an Endora, she can give you Endora Key Fragments through connecting to Gaia. This holds true for all Eidolons, even the paid ones. This however only happens rarely, though since you can get plenty of good things from connecting to Gaia, it is always recommended to use that feature.

When I state Cesela "occurs rarely" it is because there is a chance, that instead of getting a Summong Stone (or in my case an Unidentified Radiant Summoning Stone; level 60 chest from the Temple of Eidolons) you do not recieve a stone, and instead a small scene unfolds with rabbits and Cesela saying some lines (while hidden from sight), The rabbits soon disappear then Ceseka shows up to fight like just like the rest of the Eidolons from the Temple Eidolons.

Guild Hall Spawns refer to how an Eidolon spawns at the Guild hall during a set schedule (I suggest referring to Port Skandia timers) Guild Hall Summons refers to using 100 Eidolon Energy to summon Eidolons at the guild hall. Please also note that Eidolon Spawns go by a boss loot system (which charms work on). While the Summons do not, the Guild Hall Summon system uses the mail system to deliver a special treasure chest which you open for your chance at the same rewards as you would have beaten the boss version (simply put; same rewards, but Treasure Charms will not work)

Limited time Special events refers to various promotions that Aeria Games does. This ranges from things such as the Eidolon blessing event, montlhy tiered spender rewards, and bonus monthly tiered spender rewards

Although you can recieve just about any Eidolon's key fragments or full key from the Paragon Table. It depends on its availability on the paragon table, which changes a few times a week. There is also the fact that getting the specific item you want from the paragon table would require either tremendous amounts of luck, or a huge wallet of disposable income, be it from buying AP or having the piles of ruby coins.

Information on the "Eidolon Sanctuary" or otherwise the level 70 version 5 man dungeon "Temple of Eidolons" Well be added when I have experienced it, or found more information about it. As of now the best information I can find is in patch notes from when it was released. http://aurakingdom.aeriagames.com/news/123771/patch-notes-23-04-2015

Much of the format was learned from leipei.me so alot of credit is due to them. However i took simplification a step further for ease of access (especially for steam).

I list methods of obtaining in order from what I feel is easiest to the hardest. Soul Box fusion takes at least 300 tokens for ancient Eidolon Energy (that means it would take at approx up to 6 days if you farm for tokens via dailies from Mirabelle's board)

Other information was learned from conversing with friends and personal experiences (which seem neglected or omitted from the lepei.me guide) For example, how Serena can spawn in Temple of Eidolons, or like How I am told Shirayuki can spawn at Eidolon Sanctuary (which I have not confirmed).

I added the detailed information on starter Eidolon's because some people may want to know which they really want in the long run, which is not something people really mention or know when starting out. Of course things also depend on people's preferences and class choice. However the higher places on the list are certainly harder to go wrong with.

Lastly an example of a "special limited time promotion" are things such as http://aurakingdom.aeriagames.com/news/134547/eidolon-blessing which I omitted from the rest of the Eidolons because spending money outright for Eidolons that can be obtaind with relative ease without spending cash for a "chance" to get enough fragments for a key (or the actual key) seemed far too impractical to really add.