The way I see it, Saruman's fall began with his jealousy of Gandalf.
UT states that even before the Istari were assembled the germ was planted.
At a council of the Valar, where they asked for volunteers among the Maia to go as emissaries to Middle-earth, Olórin (Gandalf) was selected by Manwë after Curumo (Saruman) was chosen by Aulë, and Alatar, sent by Oromë.
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But at that Varda looked up and said: "Not as the third;" and Curumo remembered it.
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And later, Saruman may have figured out that Gandalf had been given Narya by Círdan at the Havens, at least according to
UT.
Then, there was the fact that Galadriel had tried to put forward Gandalf as the head of the White Council, which was another sore spot for Saruman.
With that jealousy in place, Saruman over time grew less concerned with the mission to defeat Sauron than with hindering Gandalf. Once he had started down that path, the pride and anger grew, making it more and more difficult for Saruman to turn back. His pride was such that he may easily have thought himself capable of controlling the Ring for his own purpose.
I think that by the time of the War of the Ring dominance of at least the West of Middle-earth was his overriding motive, with the destruction of Gandalf and those of the Eldar who supported him as an added bonus.