You can re-watch cutscenes (with your own character) by going to an Inn and clicking "The Unending Journey" book. Inns are found at the airship area in Gridania, or the taverns in Ul'dah and Limsa Lominsa.

You can replay the story by unlocking New Game+ at Vesper Bay (outside The Waking Sands in Western Thanalan). After that, type /newgame+ to start replaying the story in chapters. You can pause the story replay whenever if you want to resume your current content.

You can also replay the story on a new character, or watch the story again on YouTube.