No one has to formally teach children the basics of their native language. Each child absorbs its rules and vocabulary, first through immersion, and then by internalizing its grammar and replicating its sounds. If the child is exposed to a different language during those same formative years, it’s almost as easy to learn that language, too.
Music is a language of its own, with dialects, and immersion is the primary way it is learned. (Reminder: I am specifically discussing music of “the West”, but this holds true for all musical culture.)
So, be careful what musical language you introduce.
In an mono- or bi-lingual household, you wouldn’t baby-talk to the kids in more languages than the family actually uses, any more than you might leave a television on all day long with cartoons in dozens of languages to babysit the kids. At least, in the latter case, they might learn the native pronunciations of more languages than their own, but it would hardly cement their native language(s) in their ears and brains. (If you grew up in an immigrant household, you might remember how you learned English, but your parents kept their own native languages untaught, to retain a method of private speech. And it remained private — you didn’t learn it at home, or learned only fragments, not the actual language.)
It works the same way for music. Not everyone might agree, but I argue for surrounding kids with music that is as unenhanced as possible, that is, close to people doing ordinary singing, playing instruments, and so forth rather than heavily modified, commercially polished, or distorted-for-effect entertainment. The problem isn’t so much the “catchy tunes” that modern children’s entertainment can produce (catchy tunes are fine) — it’s the difficulty of producing those enhanced performances themselves as beginners. Better to sing along with an adult for a playsong or nursery rhyme than to watch someone else performing, for your benefit, something you can’t join in on.
Many adults think of music as a venue of entertainment that is produced for them, not something they can do for themselves. You don’t want your kids internalizing that as “normal”. Better for them to see actual people singing/playing the catchy tune (so they can join them) than thinking it can only come to them from brightly colored cartoon characters or electronically-sculpted performances or highly-esteemed professionals. (Yes, I know this sounds like an “eat your vegetables” sort of grouch, but unrestrained sugar is a problem in music just as it is in food, and bad habits start young.)
Western music shares a common vocabulary across all of its genres. Some aspects come and go historically, but it’s quite homogenous from the perspective of a language. It doesn’t really matter where you start: traditional songs, nursery rhymes, church hymns (all of these are potentially accessible to beginning performers or sing-alongs), as well as much of classical music from the Renaissance forward, especially instrumental solos or accompanied songs where the primary tune is more obvious, and various popular genres such as the blues. There’s a reason so many teenagers start with a guitar and a favorite singer in mind — they’ve listened to so much of that music that they’ve internalized what it should sound like, even if they can’t reproduce it (yet).
Particular musical genres are a bit like dialects — they all have their own internal rules overlaid on the foundation they share with other (Western) music.
If nothing else, remember that music is a language you can learn to speak, not just something you listen to. And immerse yourself (and your kids) in it so that they can feel at home in the basics, and develop a taste for particular genres or styles that they might build on for themselves later, once they have the basic tools in their ears and the habits in their fingers and voice.