The moon landings were real.
In-universe, the argument between Coop and the teacher arises because his daughter brings in a textbook which evidently contains high quality images of the moon-landings. This conflicts with the new teaching that the moon landings were a hoax, a claim that is (ironically) itself a hoax, designed to ensure that Americans aren't disappointed by the government's plans to lower people's educational expectations (e.g. we need more farmers and fewer engineers).
There's a little more info about Coop's thought process in the film's official novelisation:
“She brought this to school,” she said. “To show the other kids the
section on lunar landings…”
“Yeah,” he said, recognizing it. “It’s one of my old textbooks. She
likes the pictures.”
“This is an old federal textbook,” Miss Hanley said. “We’ve replaced
them with corrected versions.”
“You don’t believe we went to the moon?” Sure, he was aware that there
had always been a fringe element—crazies who held to that cock-eyed
nonsense. But a teacher? How could anyone with half a mind peddle that
Am I that out of touch? he wondered. Has it really gotten that bad?
He guessed he was, and that it had. He didn’t pay much attention to
what little news there was, because he had long ago realized it was
really mostly propaganda. But he hadn’t realized they had gone so far
as to rewrite the freaking textbooks.
“Okay,” Okafor said, leaning forward. “Well, right now the world
doesn’t need more engineers. We didn’t run out of planes, or
television sets. We ran out of food.”
Cooper sat back in the chair, feeling the steam leak out of him.
“The world needs farmers,” Okafor continued, with a smile that was
probably meant to be benign but just felt patronizing. “Good farmers,
like you. And Tom. We’re a caretaker generation. And things are
getting better. Maybe your grandchildren—”
Out-of-universe, director Christopher Nolan addressed this point in an interview with Slashfilm:
But we don’t think of it in those terms. We think of ourselves as
being the most magnificent, amazing universe ever and if we wanna go
back to the Moon, sure, we could. It’s like no, those guys are all
dead or retired. We’re not going back to the Moon. And if we wanted
to, we’d have to spend billions of dollars and it would take years and
years and years. We’re just done. We’re not doing that. We’re out of
that business. And so people don’t think in those terms. We had to
set the movie in the future in which that was abundantly clear.
and writer Jonathan Nolan also spoke to this issue in an interview with comingsoon.net where he makes it abundantly clear that Interstellar is set in a future where we went to the moon and then stopped:
If you look at the last 2,000 years and human civilization, things
come and go. Just because we went to the moon doesn’t mean we’re going
back to the moon. So the film had to be set in that kind of agrarian
future to hammer that home.