Just why did Galaxia set off on a quest to take over the galaxy?
Well,
first it depends on if you are looking at the anime or manga.
In the anime, Sailor Galaxia, under the control of Chaos, felt that she was the only one who could possibly be worth controlling it. After all she was the legendary soldier, the most powerful soldier in the galaxy. It was her very right to control it for herself. She was the one who saved the galaxy from Chaos in the first place. Everything else? Weak compared to her power and thus should be under her rule. At least that is the twisted view-point she picked up after being heavily corrupted by the entity known as Chaos.
In the manga she was looking for a star worth her power and felt the Galaxy Cauldron (the birth place of stars) was the only thing even close to being deserving. The Starlights mention that she maybe wanting to recreate the galaxy in her image. She was looking for something, something to obviously fill whatever void she was feeling and felt that power was the only way.
In both instances she feels that she is above everyone else, a god-like being, and therefore the galaxy should be her's.
The musicals show her as having the same motivations as the anime, with a little mix of the manga here and there. She still wanted complete control and looked down upon anyone who opposed the idea.