In my head canon, I imagine them speaking an invented language called Standard, short for Federation Standard. It’s like a super language, easily learnable, extremely descriptive and adaptable, and pronounceable by nearly every species with vocal abilities.
I wonder if they can actually learn Linguacode too, the computer or mathematical language(?) mentioned in TVH that, I think, is decipherable by any intelligent species.
Then they learn the numerical computer language we see on all Federation computer screens.
Then the main language of their memberworld. For us, it would be Anglish, the American English language of the future, which could also include plenty of Mandarin in it and other future influences.
Then maybe a bit of whatever language was popular in the nation-state you grew up in (i.e. Chekov’s Russian or Picard’s French). In fact, I wonder if earlier generations (like Robert April’s) spoke Standard, then Anglish, as their primary languages, but, with the UT becoming more ubiquitous, later generations (like Picard’s) could skip the Anglish and do Standard, then French, as their primary languages.
Then finally they might also pick up a “foreign” language. Either one popular on many worlds inside the Federation (i.e. “Andorii,” or whatever the “Anglish” version of Andorianese is), or a popular one from outside the Federation (i.e. Klingonese).
That’s a lot of languages, but it’s also DA FUTUR(!), in which they’ll probably be a lot smarter than us anyway. Look at how quickly they were abIe to bring Uhura back up to speed when Nomad(?) erased her entire memory. Also, in the real world, we may all have chips in our brains in a hundred years and be able to speak hundreds of languages with barely a thought.