The novelization indicates that Anakin actually isn't terribly interested in searching for Plagueis' apprentice directly; his priority is finding the quickest route to saving Padmé, which in this case (he believes) is learning this technique directly from the Force. There's a brief scene in the novelization, which takes place shortly after the exchange you cite, that makes this clear:
Anakin fired thrusters and slid the speeder expertly into traffic, angling toward the Jedi Temple, because that part - the part about spending the night at the Temple - was the part that wasn't a lie.
The lie was that he was going to rest. That he was going to even try. How could he rest when every time he closed his eyes he could see her screaming on the birthing table?
Now the Council's insult burned hotter than ever; he even had a name, a story, a place to start - but how could he explain to the archives Master why he needed to research a Sith legend of immortality?
Yet maybe he didn't need the archives after all.
The Temple was still the greatest nexus of Force energy on the planet, perhaps even the galaxy, and it was unquestionably the best place in the galaxy for intense, focused meditation. He had much he needed the Force to teach him, and a very short time to learn.
Revenge of the Sith Official Novelization Chapter 12: "Not From a Jedi"
We can argue about whether or not he's right, but it's not a wholly indefensible position. Finding Plagueis' apprentice involves hunting down a possibly-mythical figure, based only on a single name, derived from a tale he clearly isn't supposed to know, when the best sources of information are people who would absolutely kill this person the moment they find them1.
Two other important things to note, which the film doesn't make clear, are that:
The timeline of this movie is very short. Anakin learns that Palpatine was Plagueis' apprentice only a day, maybe two, after this conversation; although the movie only heavily implies the connection between Palpatine and Plagueis, the novelization is quite explicit about it:
"Darth Plagueis was real."
Anakin could force out only a strangled whisper. "Real . . . ?"
"Darth Plagueis was my Master. He taught me the key to his power," the shadow said, dryly matter-of-fact, "before I killed him."
Revenge of the Sith Official Novelization Chapter 15: "Death on Utapau"
Anakin was being kept pretty busy during this period. In the novelization, he comments on this more than once:
"Me?" Couldn't everyone leave him alone for day? For even a few hours?
Revenge of the Sith Official Novelization Chapter 14: "Free Fall in the Dark"
"She couldn't-" Anakin pressed a hand to his forehead; his dizziness was getting worse. When had he last eaten? He couldn't remember. It might have been before the last time he'd slept. "She could never..."
Revenge of the Sith Official Novelization Chapter 15: "Death on Utapau"
Even if he had been so inclined, he didn't really have time to do much detective work before Palpatine's dramatic reveal.
1 This is something Palpatine brings up in the novelization's version of the Opera scene, which may have also discouraged Anakin from searching for the apprentice directly:
"How would I find him?"
"I'm sure I couldn't say. You could ask your friends on the Jedi Council, I suppose-but of course, if they ever found him they'd kill him on the spot. Not as punishment for any crime, you understand. Innocence is irrelevant to the Jedi. They would kill him simply for being Sith, and his knowledge would die with him."
Revenge of the Sith Official Novelization Chapter 12: "Not From a Jedi"