People have already mentioned some of my favorites, but let me mention a new one. There’s a movie showing in many movie theaters now called Jules. It’s sot of a strange variant on E.T.. The alien reveals itself to three old people instead of three kids, and they have to protect it from certain government agents trying to find it.
I have always liked Puppet Show by Fredric Brown. A short story. Here it is in its entirety:
Yep, that’s the one.
I came to post that one. Excellent story and a real poser of a question if we ever do meet an alien civilization out among the stars…
Another of my favorites; New Folks Home by Clifford Simak, about an aging law professor and a very interesting house that leads to a new job…
Wait, was that the one where one of our ships and an alien ship are both investigating the same… something (supernova remnant?) at the same time? And we’re both, like, 99% sure that the other is friendly, but that 1% is still too uncomfortably large? IIRC, that was one where the communication issue was glossed over, but it was a very good puzzle story.
@Northern_Piper , the other one was “Does a bee care?”.
A lot of good stories have already been mentioned. I’ll add “Story of Your Life” by Ted Chiang to the list.
Oh, there’s also the episode of The Twilight Zone called “To Serve Man”.
Yes, that’s the one.
Oh very nice. Thank you. 
I would also nominate The Screwfly Solution.
Predator
Alien
Who Goes There?
I’ve always liked how the characters is This Island Earth figured out how to interpret the mysterious instructions to build an interocitor. Contact used the same idea: how would earthlings figure out how extraterrestrials would communicate with us, and the answer is “math”.
Written back when bubble memory was assumed to be the next big technological leap:
Summary
“When are we going to receive the real benefits of your technological superiority? When are you going to unlock the final secrets of the atom? When are you going to free mankind from drudgery?”
“What do you mean, ‘technological superiority’?” asked Hurv.
“There must be scientific wonders beyond our imagining aboard your mother ships.”
“Not so’s you’d notice. We’re not even so advanced as you people here on Earth. We’ve learned all sorts of wonderful things since we’ve been here.”
“What?” Luis couldn’t imagine what Hurv was trying to say.
“We don’t have anything like your astonishing bubble memories or silicon chips. We never invented anything comparable to the transistor, even. You know why the mother ships are so big?”
“My God.”
“That’s right,” said Hurv, “vacuum tubes. All our spacecraft operate on vacuum tubes. They take up a hell of a lot of space. And they burn out. Do you know how long it takes to find the goddamn tube when it burns out? Remember how people used to take bags of vacuum tubes from their television sets down to the drugstore to use the tube tester? Think of doing that with something the size of our mother ships. And we can’t just zip off into space when we feel like it. We have to let a mother ship warm up first. You have to turn the key and let the thing warm up for a couple of minutes, then you can zip off into space. It’s a goddamn pain in the neck.”
That one, I never really cared for.
Written by Frederik Pohl.
I’m going to have to look up some of these.
I’ll nitpick that and point out that “Darmok” wasn’t technically a first contact – the Federation had been in contact with the Tamarians on previous occasions, but failed to establish any sort of relations with them because they were unable to communicate with each other.
Vonda N. McIntyre’s “Starfarers”, all four books.
I’m not sure it’s my favorite, but I enjoy “A Call to Arms” by Alan Dean Foster. It’s largely from the alien perspective, but is interspersed with some chapters from the human POV, especially later in the story. What I like about it is that it is the inverse of the typical Star Trek / MCU / Multiverse trope: Humans are very likely the most physically powerful beings in the civilized universe. Stronger, faster, tougher, more physically adaptable than any other known sentient species.
Just a fun switch from the norm we see elsewhere. Of course, we’re different in other ways, but more would be in spoilervision.
We’re barely sentient by most standards, because ever single other known sentient abandoned violent conflict very early in their development, and for the matter, their conflicts were on vastly lower scale. Which makes almost all species incredibly inept at any sort of conflict - which is a shame because there is a species that sees it as their goal to unify all sentient life in the galaxy, and they have ways, even if they too are utterly inept at combat. So when one side finds Earth…
Generally though, I like stories where we see just how badly first contact can go, and just how easy it is to project our own assumptions onto others. CJ Cherryh’s Foreigner series probably being my favorite (well, the core trilogy at least).
This was going to be my suggestion. IIRC it didn’t do well at the box office because it was wrongly marketed as an Independence Day style spectacular when it really isn’t.