Luke tells Rey that
there are three lessons he will teach her as he trains her as a Jedi
What are the lessons?
He's not training her to be a Jedi. He's teaching about the Force, and explaining why he believes the Jedi Order should die. I just saw the movie, and I only remember two.
The first lesson is about the Force. Rey thinks that it is a power that the Jedi have. Luke shows her that it is not some people's special power, but an energy that connects all things. He also claims that it has a natural balance between light and dark, and that it doesn't need the Jedi to maintain it.
The second lesson is about failure. The Jedi Order failed to notice a Sith Lord rising in power until it was too late. Obi-Wan failed in his training of Anakin. Luke failed in his training of Ben Solo, and one moment of weakness cemented his turn to the dark side. Because of these failings, he believes that the galaxy would be better off without them. (Events later in the movie suggest that he changed his mind)
I don't remember a third lesson. It may be that he discovered Rey's connection to Kylo, and she left, before he could give his third lesson. (And this could be a deliberate parallel to The Empire Strikes Back, when Luke left to confront Vader before finishing his training.)
The third lesson was deleted from the movie, because it contained Luke acting like a total jerk (emphasis mine)
One of the biggest deleted scenes involves another lesson Luke Skywalker gives to Rey, taking place chronologically after they talk inside the cave where Luke mentions Darth Sidious. Rey notices boats arriving at the island and there appears to be a big fire from where they’re landing. Luke tells her that it is a group of bandits who regularly come back to the island to plunder and kill the caretakers. Rey is very concerned and wants to help them, but Luke tells her that if you help them now, the raiders will come back stronger and it will make things worse in the future. He asks Rey if she is always going to be here to protect them, saying that a true Jedi Knight would do nothing and would only act to maintain balance, even when people get hurt.
Rey, furious at his reasoning, ignites her lightsaber and runs really fast, a Force-powered run that we glimpsed in one of the featurettes about the making of the film (see the screenshot at the top of this article). She runs over rocks on shallow water and bursts through a door with her saber into the village square ready for battle. Luke yells for her to wait, but she doesn’t stop.
She is surprised to learn that it’s not a raiding party, but an actual party, with caretakers celebrating and swinging glow sticks.
Rey is mad that Luke lied to her and she confronts him. He admits that he’s sorry, but that she ran so fast and he couldn’t stop her. Rey says that she thought they were in danger and tried to do something. Luke responds, seriously this time, that that’s exactly what the resistance needs – not some old husk of a failed religion. He was again trying to teach her a lesson. Rey cries, explaining that her real friends are really dying and “that old legend of Luke Skywalker that you hate so much, I believed in it.” Luke is in shock. He realizes that he pushed her too far. Rey tells him she was wrong about believing in him and storms away.
She left before he taught her the third lesson. But in reality he did teach her 3 things; just like Yoda mentions, the third lesson was his failure to Ben Solo. The second wasn't so much about failure in my opinion as it was about reality, the difference between what's real and what's a legend, showing her that he was in the end just a man. That's my understanding at least.
But being honest, I think their relationship is much more about Rey bringing him back than he teaching her the ways of the Jedi.