First of all: Yes, you probably should watch at least The Two Towers before reading much further: that movie is all about Saruman's plans and schemes.
Having said that, there's a combination of things that have driven Saruman to act the way he does in the Lord of the Rings trilogy. The first thing you need to realize is just how long a gap there was between the two trilogies. When Bilbo went on his adventure in The Hobbit, he was about 50; at the beginning of Fellowship of the Ring, he's celebrating his 111th birthday - 61 years later.
During that time, a lot has happened to Saruman; the main things he's done are:
- He's started using the Palantir to spy on Sauron; this is the crystal ball in his chambers in Orthanc, the one that freaks Gandalf out when he sees it. Gandalf is worried about the effect of using those stones, and says so, right off the bat; later on we find out that Sauron not only has one, but that he can use it to extend his influence a bit and corrupt the mind of its user (and Saruman isn't the only victim).
- He's taken up ringcraft; I don't remember if this is ever really explored in detail in any of the editions of the movie, but Saruman has been trying to recreate magic rings of his own, attempting to rediscover what Sauron had managed to do with his Rings of Power. Having the One Ring in his possession would be a huge boon to his research -- or so he tells himself. Really, like everyone else who desires the ring, he's basically fooling himself, and ultimately playing into Sauron's hand. But knowing what Gandalf wants to do with it, Saruman certainly doesn't want him to succeed.
- He's suffering a tiny bit of a nervous breakdown over Sauron's return. This partly goes back to point #1, Sauron getting into Saruman's head via the Palantir, but also the fact that Saruman seems to genuinely believe that Sauron is strong enough to defeat all of Middle-earth. Saruman and the other wizards were sent to Middle-earth specifically to fight Sauron, but Saruman has apparently decided that such a job is hopeless. The only way to survive what he knows is coming, in his mind, is to ally with Sauron and hope they don't get betrayed.
- Finally, there has always been a tiny bit of tension between the two. The big issue is the fact that Saruman is, nominally, the head of the Wizards, so Gandalf is technically his subordinate. However, Gandalf, by all accounts, seems to be the "better" wizard. In fact, Gandalf was the first choice to have Saruman's position, and turned it down. Gandalf's also the one the Elves and Rangers and others turn to when they need help. Additionally, More specifically, he's the one that was given one of the Three Elven Rings of Power (I think you'll find that out at the very very end of Return of the King). Thus, Saruman is a bit jealous of Gandalf, and probably worried that Gandalf will try to take Saruman's position (and, in Saruman's mind, probably The One Ring) for himself.
All of these things, taken together, make Saruman decide that he and Gandalf do not have the same goals. At first, he doesn't try to actually hurt Gandalf, showing there's probably a modicum of respect left between them. But, he has no intention of letting Gandalf destroy the ring, and tries to take him out of play while Saruman's agents search for the ring.