So, I've always wondered if Sidious ever taught ANYTHING to Vader. I've been reading Lords of the Sith and that combined with the movies leads me to believe that at most, Sidious only ever gives him tiny bits of insight about how a Sith Lord thinks and operates, but that seems completely minuscule when compared to the "You’ll be able to prevent loved ones from dying" promise he made to Anakin.
Sidious made no such promise
Sidious never actually promised Anakin that he'd teach him how to cheat death - he just claimed that the dark side was a path to such abilities. Here is the relevent scene:
PALPATINE: (continuing) Did you ever hear the tragedy of Darth Plagueis "the wise"?
PALPATINE: I thought not. It's not a story the Jedi would tell you. It's a Sith legend. Darth Plagueis was a Dark Lord of the Sith, so powerful and so wise he could use the Force to influence the midi-chlorians to create life ... He had such a knowledge of the dark side that he could even keep the ones he cared about from dying.
ANAKIN: He could actually save people from death?
PALPATINE: The dark side of the Force is a pathway to many abilities some consider to be unnatural.
ANAKIN: What happened to him?
PALPATINE: He became so powerful . . . the only thing he was afraid of was losing his power, which eventually, of course, he did. Unfortunately, he taught his apprentice everything he knew, then his apprentice killed him in his sleep. (smiles) Plagueis never saw it coming. It's ironic he could save others from death, but not himself.
ANAKIN: Is it possible to learn this power?
PALPATINE: Not from a Jedi.
And later, when all is out in the open, it's clearly stated:
ANAKIN: Just help me save Padme's life. I can't live without her. I won't let her die. I want the power to stop death.
PALPATINE: To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret.
ANAKIN kneels before PALPATINE.
ANAKIN: I pledge myself to your teachings. To the ways of the Sith.
That was Sidious' promise, then - to work with Vader to find the secret.
But remember what happened almost immediately after that - the next time Sidious and Vader met, Padme was already dead. Vader no longer had any reason to search for the secret of saving people from death, because there was nobody he cared about any longer. He was full of rage and hatred for everyone, up to and including himself.
So, what did Sidious teach Vader?
If not the secret of keeping people from death, then what teachings DID Sidious have for Vader? Likely the same things he taught Maul: the history and the ways of the Sith, the uses of the dark side, the power that comes from rage and hatred, as well as tactics and strategy.
It's unlikely that Sidious regularly sat down with Vader for tutoring sessions where Sidious wrote on the blackboard and Vader brought him an apple every morning; rather he taught through object lessons and experience. Vader's tasks of hunting down the Jedi who had survived Order 66, supervising the Inquisitors, supporting the Imperial military...these were all lessons. It's also likely why Leia expected to find someone "holding Vader's leash" - he wasn't (yet) known for operating independently as a commander in the Imperial military.
In fairness to Palpatine1, he did intend to continue Plagueis' research with Vader; it's just that the business of Empire-building kept getting in his way; this is discussed in the canon novel Tarkin (bold is my emphasis, italic from the text):
The sooner the [Death Star] is completed, the sooner you [Vader] and I [Sidious] can devote ourselves to more pressing matters - matters only you and I can investigate and that have little to do with the Empire."
Save for Sidious, no sentient being in close to five thousand years had set foot in the [Sith shrine beneath the Imperial Palace]. The room's excavation and restoration had been carried out by machines under the supervision of 11-4D. Even Vader was unaware of the shrine's existence. But it was here that they would one day work together the way Sidious and Plagueis had to coax from the dark side its final secrets.
Tarkin Chapter 9: "As Above, So Below"
In general, however, you appear to be largely right: there doesn't seem to be much formal instruction in their relationship; if there is any, we don't see it. Sidious' great talent is maneuvering Vader into situations where he can "learn on the job". This is, broadly speaking, the point of Lords of the Sith, the first arc of the Marvel Darth Vader comic series, and Tarkin to a lesser extent.
1 Boy is that a sentence I never thought I'd write...
Why would you assume a Sith lord would tell the truth? Or any Sith, lord or not? I mean, presumably they tell the truth sometimes, because otherwise they won't ever eat what they want to or, you know, get a space army that salutes the way they want them too.
So I suppose Sith do tell the truth, but only when it benefits them. But only the most superficial benefit, not that warm feeling that you get for living a good honest life and not having to constantly wonder if people are lying to you - no, just the benefit of getting your eggs sunny side up and an army that salutes with their left hand. Which is a good benefit to be sure, but it's best not to ponder too long on the depth of what might benefit someone.
Anyway, I don't think Vader paid money or entered a legally binding contract to learn things - although maybe the deal might have been an interesting commentary of Trump University, albeit years ahead of its time? But no, I think Vader just took on the role because he, uh, didn't have much doing and didn't want friends and wanted to rule a galaxy for reasons that are never made clear - again, Sith aren't the type who navel gaze, they really are doers.