Youre asking about the important story points from the first 20 or so levels? I can give a short overview. In addition, if you retire to an inn room(any city) theres an book there called the "Unending Journey" that allows you to rewatch cutscenes. Normally i would use spoiler tags, but since this is just the first 20 levels it doesnt really feel spoilery.
The surprises happen later.
So, 5 years ago there was during a big fight between the alliance of the Eorzean nations and the invading Garlean Empire. At the end of it, the moon crashed down and bahamut broke out of it. He trashed the place. Louisoix(a revered hero) used some kinda big magic spell to win the fight. He died in the process. There were Warriors of Light who helped, but no one can remember them for some reason. And no one knows what happened to them.
Fast forward to today. You are a traveler who arrived in Gridania. Some stuff happened - i havent played the Gridanian story - but you met Yda and Papalymo(the lala and hyur you mentioned). You proved yourself somehow to be a good hero. So(im guessing) the elder seedseer sent you as an envoy to the other 2 cities. They were doing memorials for the people who died when the moon crashed and trying to rally the peoples spirits. You met a couple of whiny twins at all 3 ceremonies. While visiting each of the towns you were asked you to look into a local problem(3 dungeons). Around here you were introduced to the Scions of the Seventh Dawn - a group trying to stop these big calamities from happening - including Yda and Papalymo who watches over Gridania for them(also those whiny twins who are the grandchildren of Louisoix). You joined them.
The next major story point was when you heard that the Amalj'aa were either about to or just did summon their god Ifrit to burn the world. You were sent to investigate. Then somehow you were captured. Ifrit tried to brainwash you, but you have a special power called the Echo that made you immune. You probly havent learned much about the Echo yet. So since you were immune to brainwashing, you fought Ifrit and won. Afterwords because of your exemplary heroism you were asked to join one of the three grand companies, so you chose one. And finally, the Scions have asked you to investigate the sylphs. The Gridanians are worried that they plan to summon Ramuh, so you were sent to befriend the non-crazy ones and see whats going on. And now youre in part of a long series of quests(your words) to figure that out.
Youve also seen hints that theres more going on in some of these situations then meets the eye. Some mysterious figures here and there, some clues that dont make sense. But thats the short version of where youre at. Someone else might be able to give the Gridanian story of the first 15 levels. And if you want a refresher, you can watch any of the old scenes to see what happend.