GUIDE
(Note, this is not a full guide, this is just how I completed the event.)
Part 1 - lvl 12
Travel to Limsa Lominsa Lower Decks, and speak to Petyr Winsome to initiate the first 'mini quest'. He will tell you of a strange adventurer that he would like to report on, and he wants you to seek them out.
For this part, I did the 'Sister Crustacean' FATE in Middle La Noscea. You can find it at around X17:Y9 on the map! It will respawn every 7-10 minutes so just hang around there. NOTE: You must complete the FATE with a Silver of Gold medal and Lightning must show up (she will only show up for 1 of 2 respawns. So if you don't get her on your first try, she'll show up for the second.) As a prize, you'll get 2 Otherworldly Parcels which you present to Petyr.
Part 2 - lvl 26
Now go to Ul'Dah and travel towards the Chocobokeep to find another reporter called Dural Tharal. He has also heard of Lightning and wants some info. Again, you will have to FATE.
I went north of Highbridge in Eastern Thanalan around X26:Y18. The FATE is called 'Lightning Strikes: Armoured Beast' and you will know it is being triggered when the NPC's of the area all rush toward the source. Lightning will show up again to help you take down the mecha-behemoth creature! After completely this you get a Strange Otherworldly Parcel which you give to Dural Tharal.
Part 3 - lvl 38
Travel to New Gridania and then take the Aethernet to the Amphitheatre and speak to Kipih Jakkya. She sends you to Coerthas Central Highlands for another Lightning sighting!
Go to Camp Dragonhead and head towards Whitebrim. The FATE I did spawned at X16:Y20. This time you fight a Dreadnought in 'Lightning Strikes - Dread on Arrival' and you'll receive a Simple Otherworldly Parcel. Take it back to Kipih!
Part 4 - lvl 45
After a short cutscene with Kipih and Lightning, you are sent over to Mor Dhona to help out Lightning once more! So head there for the last two FATEs of the event!
Travle to Revenants Toll, I found the FATEs at the location of X26:Y11 on the map! The first FATE 'Anubys Assault' will spawn first - so complete this one and then the next FATE will pop immediately after 'On Wings of Chaos'. During the last FATE Lightning will summon up her Final Fantasy XIII Odin which is a great sight to see!
After you've won the FATEs, travel back to the Mih'Khetto Amphitheatre to complete the event! There is a cute cutscene that is worth watching, fan of Final Fantasy XIII or not!
ITEMS GAINED
For each part of the event you do, you will gain a new piece of Lightning or Snow's outfit (depending on gender). In Part 1 you will get the gloves, in Part 2 the skirt/pants, and in Part 3 you will get the boots. After completing all 4 parts, you will get the coat and the hairstyle of Lightning or Snow will now be available from the Aesthetician. You can also purchase Final Fantasy XIII style weapons for each class from Syngiwyb (stood next to Petyr Winsome) in exchange for Otherworldy Parcels which you obtain in the La Noscea Lightning FATEs.
This was a great event to take part in. It took very little time, but was so exciting! It was great to see Lightning in Eorzea and to pick up some cute new items! What I also love is how many people are dressed up as Lightning in Eorzea right now (myself incuded!)