Keep in mind that you can rewatch cutscenes from within your room at any Inn but, if you'd rather just read the Wiki of the main story, here it is:
http://finalfantasy.wikia.com/wiki/Seventh_Umbral_Era

It contains all of the info, along with optional content that has appeared along the way. My favorite part if the photo of Dalamud and its increasing size/proximity with each patch!

http://images.wikia.com/finalfantasy/images/8/87/FFXIV_Dalamud_120.jpg