What exactly is this Lightning thing?

[Enkidoh]

The reason why the this event increases in level requirement with each successive quest is because they all involve specific FATEs to be completed which take place in areas of those levels (the last quest for instance, has a FATE that occurs in Mor Dhona, a level 45-50 area). Having said that, the first quest is easy enough, as all you have to do is complete either the FATEs 'Sister Crustacean' in Middle La Noscea, or 'Tender Buttons' in Western La Noscea. Doing so will gain you two key items.

One of these items is required to complete the quest, gaining you one part of Lightning or Snow's costume (the gloves if I remember rightly), while the second item can be redeemed with the vendor next to the quest npc for one FFXIII-themed weapon. Note that those FATEs can be done repeatedly, thus allowing you to gain all the weapons (you can only do the quest once though).

The remaining three event quests will each send you to a special FFXIII-themed FATE, which again completing it will gain you a new key item, which upon trading to the npc gives another piece of FFXIII armour, until finally, you'll complete the full set.

Hopefully that hasn't been too confusing for you.