How did Luke Skywalker learn to communicate with Astromech droids? How did he learn the language whilst living on Tatooine?

I don't think he understood binary during a New Hope. C3PO has to translate for R2 when they first meet.

Also he reads what R2 says via a screen in his X wing in ESB. I don’t think he understands binary until at least RotJ. Maybe he got bored in mechanical hand physical therapy and also studied droidspeak.
- I doubt he got bored. That mechanical hand probably felt like someone else's until the nerves fully adjusted.
He understands him perfectly in ESB.
- I think it's implied that having spent enough time with R2, he eventually picks it up. I know that according to the lore, all X-wings are equipped with a translator, so that pilots can understand their astromechs. Given enough time flying together, it makes sense that Luke would need the translator less and less.
  Not to mention it just helps move the story along, considering all the scenes they spend alone together in ESB, without C3PO. In a New Hope, C3PO is always around to translate when R2 has something to say.
- Usually through the text translation on the X-Wing screen. Once on Dagobah, he's picking up the same context clues in tone that the audience is.

Duolingo

Droidolingo

I could never understand why they didn't just throw a speech module into R2-D2.

Cause all those beeps were cussing.
Because back in the 70s, computerized speech still seamed like something extremely difficult and very sci-fi. That's why only droids who really need it have a speech module.
Nowadays every cheapo smartphone has speech synthesis capabilities, but back then that was very sci-fi.
Btw, that's why a ton of modern sci-fi is retro-sci-fi (basically steam punk but with 70s computers), because real tech has surpassed sci-fi in many ways.
- "Are you ready for science fiction? Well. I have this robot in the story....and he can talk! He can do basic calculations, AND speak the answers!!! ISN'T THAT WILD???"
  producer pulls out iPhone
  "Hey Siri, whats 4,684,854,853 divided by 7?"
  "669,264,979"
  And this joke exchange would have been more impressive if everyone hadn't known that I'm on a cell phone, with a built in calculator.
  Seriously, any cell phone today would have been called INSANE in the 1970s.
  I mean think about it. In the late 1960s Maxwell Smart talked into a shoe phone. And it was considered crazy high tech. So much so that as a kid in the late 80s, it was STILL crazy high tech to just have a phone. Just out and about.
  I'm now old enough to know that technically cell phones existed at that time. But the fact that I was unaware they existed should serve as a stark contrast between cell phones in 1988 vs 2025. Ask any 5 year old today what a cell phone is, and they'll know. Now have them watch the original Get Smart series, and watch them get confused by why he has a cell phone inside his shoe. I would bet they wouldn't be as excited as I was when I saw how cool the shoe phone was. Today, it would be weird.
  To be fair though, the opening sequence with the doors, IS still cool as fuck.
With the way how stupid LLMs are especially with maths I now think it's actually quite logical how you wouldn't want something like that in a droid that is mainly meant to do astronomical calculations.
In the novelization of ROTJ it is also revealed that the torture droid in Jabba's palace instructs C-3PO to just answer yes or no in order to reduce him to binary because that is something protocol droids are ill equipped for. On the other hand he speaks complex sentences to R2, I guess for much the same reason but the other way around.