Just how does Galaxia think?
Well, in the anime it was more of Chaos than it was actually Galaxia herself.
However, with her corrupted thinking she felt that all other beings were
lower than herself. She was, after all, the Legendary Soldier of the galaxy.
She held a respect for no one, the Animamates were worthless tools, easily 
disposed of.
As her good self, she loved the galaxy and wanted to protect it. She understood her mistakes -- sending her Light of Hope out as a way to give the galaxy as last chance, truly a great warrior. However, this is about as much of her personality as we get to see. Unlike the heroes of the series, the villains did not get a chance at much character development. We only learn about Galaxia's true past in the finial few episodes. We get to see more of her corrupted verion's personality due to her interactions and treatments of the Sailor Animamates.
In the manga, her personality was slightly the same, but still a little different. Think of Anime Nehellenia and you sort of have a glimpse into her manga self. She still felt every other being in the galaxy was lower than her, but it is uncertain whether she was the true galaxy soldier or not. She was sarcastic and manipulative, often getting others to do her bidding quite easily, though at times, she could even be rather polite.
She knew Sailormoon had more power than her, and respected her only for that fact, she hated the moon soldier for her ideals and beliefs. To Galaxia love was worthless, empty, useless, friendship was even worse -- you cannot trust others, power is the only thing worth trusting. She couldn't understand why people put so much faith in false ideas -- perhaps this is what caused her downfall.
In the musicals she is combination of both worlds. She is still ruthless and cruel, but in a few of them we get to see a glimpse of her good self as seen in the anime. Her treatment of the Shadow Galactica members is about the same, as she would not bat an eye at disposing of someone she believed to be a failure. She does seem to be a bit more manipulative as she convinces characters such as Queen Beryl to join her side.