So for fucking what exactly?
I might be repeating what someone else has said, but fuck it, I don't have time to read the whole thread right now. So Phos's most defining characteristic is the ability to change, which is highly unusual as the gems are characterized by their rigidity. They are slaves to tradition, even when those traditions cause harm or are nonsensical. This difference is the first hint that Phos doesn't fit into gem society, which manifests in other ways that contribute to Phos feeling useless to their society. Her motivations and desires change too much to be effective at what the others do. She is much like an actual human in this way.
We also know from sensei that Phos has always had a kind heart. So her early motivations are 1) finding a fulfilling purpose, and 2) improving the lives of the gems. 1) is difficult because, much like humans, Phos's fickle nature's means that what is satisfying right now might change later. 2) is synonymous with wanting to resolve the Lunarian conflict because it is the only real source of suffering for the gems, and unlike the others who can only keep doing what they do, she has the potential to actually bring change - to break the (karmic) cycle.
Over time, Phos changes physically, psychologically, and philosophically, because that is her nature. Since these things all drive, well, everything about us, that means her motivations also change. The big turning point is the introduction of Lapis's inclusions, and thus part of her curious and skeptical personality. Now Phos's motivations turn towards finding the truth. Why? Well, now that's part of her nature, because she changed. She is still part Phos, so you can argue that this is driven by the thought that finding the truth will ultimately lead to a better gem society, but we can't truly know what her real motivation was. That's for the better, I think; it makes Phos a very interesting and realistic character, multifaceted unlike other gems.