Rebel: Freedom.
Even though the Republic has similar goals, they still frown upon certain people like the Sith and bounty hunters and what not. The Rebel Alliance wants to bring everyone together (kind of like the American Revolution: Freedom of speech and religion and what not).
The Sith Empire get in the way of that because they want to destroy everyone who opposes them, which is us because in order for us to reach out goal we have to bring down the Empire's dictatorship.
The Imperial Remnant seek to bring order to the chaos. I am thinking of trying to start a small relationship with them and eventually make them allies.
The Yuuzhan Vong, well I don't really know much about them. But I think they oppose the empire and could be allies.
The Omega Pyre is essentially the Talon Mercenaries of Star Wars. They don't care who wins as long as they make money. This means that they would oppose the idea of everyone being together since they would make no income since everyone is at peace.
The Syndicate isn't really a problem and since they are a smaller faction, I don't worry to much about them.
So in recap, we fight for freedom. But true freedom; freedom of speech, religion, ect.