Let's see...
Earthlings: SC's Earthlings would want to expand their sphere, but probably through colonization, not conquest. (I'm talking about Star Control Earthlings here, not real ones.
)
Syreen: Same as the Earthlings for the most part, except I think they would be a little less adverse to war. (After all, they carry daggers whereever they go.
)
Mycon: We must change the Non in the name of Juffo-Wup.
Ur-Quan: Duh.
Shofixti: Feudal Japanese warlords were always jockeying for more power, and the Shofixti are based on them... so yeah, they would expand their sphere.
Spathi: Probably want to expand their sphere just so they have more advance warning before The Ultimate Evil reaches Spathiwa.
Yehat: Colonization, yes. War, no, unless the people they are to attack insulted their honor first.
Pkunk: They probably want to spread as much love as they possibly can, but without bloodshed.
Thraddash: Would be more than happy to conquer other races... if they could stop conquering themselves for a few years.
Ilwrath: Increasing the size of their empire is probably not their goal, but when your fetid gods command you do conquer someone, it happens anyway.
Androsynth: Interested in building up their empire as much as possible before the conflict with the humans that we all know was gonna happen if not for the Orz.
Orz: Umm... I dunno.
Arilou: Who knows?
Kohr-Ah: They want to be the ONLY sphere of influence.
Chenjesu: Would have no problem with colonization as far as I can see, but would never conquer others. If forced into a war, once they won they would probably set up a new government there and leave. (Like the U.S. and Japan after WW2)
Mrmnmhrm: Can't really expand, as their population can't grow. May still explore other Precursor related planets to look for a way to repair Mother Ark.
Umgah: Spread as far as they can, but without violence. Well, unless that violence was funny...
Druuge: More worlds means more capital for the Crimson Corporation.
Utwig: Like the Earthlings, peaceful expansion.
Supox: Do whatever the Utwig do.
VUX: They strike me as pretty warlike...
ZFP: Peaceful expansion would be to their liking, but that's sort of hard to do in the middle of to Ur-Quan spheres.
Am I forgetting anyone?