Obi-Wan doesn't seem to have trained Luke in this period, but Legends says Luke did read Obi-Wan's books.
From the Disney canon novel Star Wars: Heir to the Jedi:
Luke Skywalker, hero of the Battle of Yavin, has cast his lot with the rebels, lending his formidable piloting skills to whatever missions his leaders assign him. But he is haunted by his all-too-brief lessons with Obi-Wan Kenobi and the growing certainty that mastery of the Force will be his path to victory over the Empire. Adrift without Old Ben’s mentorship, determined to serve the Rebellion any way he can, Luke searches for ways to improve his skills in the Force...
THERE’S NO ONE AROUND to answer all my questions now that Ben’s gone... I thought I heard Ben’s voice a couple of times during the Battle of Yavin, but I’m wondering now if that really happened. Maybe I only thought it did; maybe that was my subconscious speaking to me— a kind of wishful thinking. He’s been silent since, and I don’t feel I can talk to anyone else about the Force.
And another Disney canon novel, Star Wars: The Weapon of a Jedi, agrees:
But he hadn’t heard Ben’s voice since— and he feared he never would again.
In the Legends novel Star Wars: Shadows of the Empire, Luke finds a book in Obi-Wan's hut and uses it to build his second lightsaber. This occurs between Empire Strikes Back and Return of the Jedi:
Luke Skywalker stood barefoot on the steelcrete platform sixty meters above the sand, looking at the taut wire. He wore plain black pants and shirt and a black leather belt. He no longer had a lightsaber, though he’d started constructing another one, using the plans he’d found in an old leather-bound book at Ben Kenobi’s.
He looked around at the inside of what had been Ben Kenobi’s home. It was a small place on the edge of the Western Dune Sea, made, as so many of the local structures were, of synstone— crushed local rock mixed into a slurry with dissolvants and cast or sprayed onto frames to harden.
Perhaps the most valuable thing was the old and intricately carved boa-wood trunk and its contents, including an ancient leather-bound book. A book that contained all manner of wondrous things for a would-be Jedi, such as plans for building a lightsaber. The thumbprint clasp on the volume had accepted Luke’s right thumb to unlock it, and once it was open, he saw the flashpacket rigged inside the cover. Had anybody tried to force the clasp, the book would have burst into flame.
Somehow, Ben had known Luke would find this book. Somehow, he had prepared it so that only he could open it safely.
According to that book, the best lightsabers used natural jewels, but there weren’t a lot of the kind he needed lying around where he could find them on Tatooine. He’d managed to collect most of the electronic and mechanical parts in Mos Eisley— power cells, controls, a high-energy reflector cup— but he had to make his own focusing jewel. Ideally, the best lightsabers also had three of those, different densities and facets, for a fully adjustable blade, but for his first attempt at building the Jedi weapon, Luke wanted to keep it as simple as possible. Even so, it was trickier than the book made it out. He was pretty sure he had the superconductor tuned right, the amplitude for the length set where it was supposed to be, and the control circuitry boards correctly installed. He couldn’t be positive until the jewel was finished, and the book didn’t mention exactly how long that took. Supposedly the furnace would shut down automatically when it was done.
Luke took a deep breath. He stood outside Ben’s house, the first stars of evening aglimmer, the moon still on the rise. The air was warm but not as scorchingly hot as it had been. He held the completed lightsaber in his right hand. He had assembled it according to the old book’s direction; everything should work.
Another Legends novel, Star Wars: The Lives and Adventures of Luke Skywalker, Obi-Wan Kenobi, and Darth Vader, makes it clear that Obi-Wan intended for Luke to find these books:
Peering past the flashpacket, Luke saw that the box contained some rectangular objects. He recognized them as books. Although he was far more familiar with datapads for information storage, he had seen enough books in his lifetime to know how what they were and how to use them. The largest book was a leather-bound volume that appeared quite ancient. Luke picked it up, and noticed that it too was sealed by a thumbclasp.
The flashpackets were a necessary precaution. I trust you will dispose of them properly.
The future of the Jedi Knights is in your hands. Read these books and use them wisely.
May the Force be with you.
Luke blinked at the words as if to confirm they were real, that he wasn’t just having a dream. The book felt suddenly heavy in his hands. He set it down carefully upon the workbench and, by the light of his glowrod, he began turning the pages. Every page was filled with handwritten text, and his heart began pounding harder as the various words and phrases caught his attention. Jedi Council... Old Republic... Battle of Naboo... Sith Lords... Jedi Temple... Separatist Movement... Battle of Geonosis... the Clone Wars...
The dull ache returned to Luke’s right wrist, and he stopped turning pages. He hadn’t found the names he was looking for, but had come to a section that contained Ben’s instructions for the construction of lightsabers. The section included numerous illustrations by Ben himself. Luke hadn’t considered the possibility of building a lightsaber. Only after he lost his lightsaber at Cloud City did he realize that he had no idea where to obtain another one, let alone how to go about making one from scratch. Now, thanks to Ben’s book, it seemed he might actually stand a good chance at replacing it.
In fact, Obi-Wan had begun compiling the books years earlier, specifically for Luke's use.
The next time Obi-Wan visited Anchorhead to obtain supplies, he found an unusual rectangular-shaped item in a junk shop. The shopkeeper was apparently unfamiliar with the item’s function, and was using it as a shelf to display a small selection of used power couplings. But Obi-Wan— now known locally as Ben— recalled handling similar objects in the collection of the Jedi Archives, and recognized the “shelf” as an ancient, leather-bound book.
Obi-Wan moved the power couplings aside and opened the book. Incredibly, only a few pages were slightly discolored, and all were blank. He had never considered writing a journal before, but suddenly realized that a journal would be a good way to preserve information about the Jedi.
Information that Luke might need someday.
Holding the book up for the shopkeeper to see, Obi-Wan said, “Do you know if this thing will burn properly?”
The shopkeeper shrugged. “Beats me what it’s good for, Ben,” he said. “But it’s yours for a credit.”
Ben did not haggle.