Is the game's story completely changing in the new version or restarting?
The main scenario is staying more or less the same, there are going to be quite a few changes to the cutscenes and the pacing, though. Certain scenes will make allusions to the Seventh Umbral Era meteor storyline.
The grand company questline will be completely new and restarted from scratch.
If the story isn't changing or restarting, is the new version of the game like an expansion of the story?
Kind of. It's a more-fleshed out version of what we had in 1.0. It will have more tutorials on how to play and get around, and more boss battles and not completable/soloable as the 1.0 version was.
Should I read more about the story of the game before the new version comes out because I don't want to spoil it.
You should read up on the Umbral Era if you want some background on the story. I made a blog post for people who want to catch up on what has been missed, but I imagine the storyline will be accessible in-game to those who are trying the game for the first time.