You "show" someone a picture. You "---" someone a song?

In English, we would usually use the word play. That covers both the act of performing the song on an instrument (perhaps while singing, if the instrument permits) and the act of playing a recording of the song. If you give an a cappella rendition, live, you would use the word sing.

Per @bye's answer, in the case of a song, you'd probably play or sing it to someone else (depending on how you cause the sound to be made). But if you wanted the band at your private party to play some particular song that you can't think of the name of, the bandleader might say:-

"You hum it and we'll play it" (just give us idea of what it sounds like; we'll probably recognise it)

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More generic verbs (which could also be used of a taste or smell as well as a sight or sound) are:-

present - to offer for observation, examination, or consideration; show or display
demonstrate - to give a demonstration

Play is good if you want something specifically for sound, but I would generally use show here. It refers to introducing or presenting new information in a variety of contexts, not strictly related to sight. For example, “He showed me a great new sandwich shop”. I find it no different than saying “I see” to metaphorically mean “I understand”.

"I will sing you a song " seems good enough

In Russian, we say "дать послушать" which means something like give to listen or give a listen (according to Google translate), like if you have some recording (MP3, CD etc) and you give that recording to someone. In English, it's similar. You could say to your friend "give a listen to this song" or "give this song a listen".

I tend to say "I'll load up that song for you", but that phrase tends to imply that the song is stored as a file on a computer, so it might not work if the recording is stored on a more old-fashioned medium.

I think the closest is "have you listen to," but it's only candid to say you've found an asymmetry in English, and native speakers will have slight awkwardness when ???ing someone a song. Possible ways this might be done:

• "I want to show you this song" somewhat acceptable still, possibly because it's understood there's no perfect way to say this
• "I want you to listen to this song" acceptable, but different color since it's not me doing something to/for you anymore.
• "I want to have you listen to this song" resolves above difficulty but is longer and more awkward that way. I think it's preferred.

As was suggested, you could use the word play. But the word show is also acceptable. The word show does not exclusively mean cause to be seen. Other meanings include:

3 : to present as a public spectacle : perform
6
    a : to point out : direct attention to
    b : conduct, usher
7 : accord, bestow
Source

Definition six (a) is what you mean when you "show" someone a song, typically. You are probably going to pull it up on YouTube or something so they can hear it for the first time. The assumption under this scenario is that they have never heard the song before and showing it to them directs their attention to it. Playing a song for someone does not necessarily imply that.

'Make mention' is a passable English phrase for "mention for you", insemmgħek. As pointed out, there's no one-English-word to translate this compound word into. When someone plays music for another, they are serenading them. However, this normally connotes a romantic, appropriate, and well received selection, unless said in jest.

I don't believe that there is a single-word English translation of the concept of "make someone listen to something." However, if we stop focusing on the looking-vs.-listening distinction, we have terms like:

• Share. For example, the Internet is full of buttons that let you share a link with your friends, meaning that you want someone else to have the same experience you had.
• Introduce. That is a form of sharing where you believe that someone does not yet know of something.
• Recommend. That is a form of sharing where you expect that someone will like the experience, because you know the other person's preferences.