It really depends on how prevalent the UT is and what is required to run it. From DS9 we know that the Ferengi use implants though the Next Gen era Federation uses comm badges according to the tech manuals. We don't ever see a pure civilian translator but we can assume that that the federation com badges translate both what the wearer is saying and what they are hearing. Before then (Ent and TOS era) UT tech is bulky and probable not in wide use without some sort of centralized service like Siri where a lightweight client relays to a powerful machine that sends back the translation. Implants tends to be very rare among Federation peoples often times called out as a differentiator since they tend to go for reconstruction or manipulation over replacement.
Outside of Starfleet it would be questionable how prevalent UT's would be. Most communities would probably settle on one or two common languages they share. Most people grow up outside of starfleet so unless their colony or community has a large starfleet presence they may or may not speak Federation Standard growing up. This could indicate why cultural accents still exists. We do see people in Starfleet learning other languages and in "Firstborn" TNG s7e21 we see colonists speaking in both Klingonese and Federation Standard for the audience. Within Starfleet everyone would either be speaking Federation Standard or their translators would be providing Federation Standard for listeners.
In the end most Federation citizens would be raised Bilingual unless they are living in a culture on the outskirts of Federation society or a community that holds it's pre-federation cultural identity. We see this with Keiko O'Brien, B'Elanna, and Chakotay where they have a working knowledge of their cultural language but would have learned Federation Standard
However, those born into Starfleet families like La Forge or Harry Kim would probably be raised with Federation Standard since it is what is coming out of everyone's translators or mouths. Unless they have a particular interest they would not learn other languages.
As for written script there is no real explanation as to how they can read languages of races they have had little to no contact with. However, characters will fairly consistently hold a tricorder up next to the screen or open it and set it next to the screen so we can assume tricorders come loaded with some sort of script translator. But as I said, this is not consistent.
Beyond Federation races we have even less to go on. As for Klingons, Romulans, or Cardassians there is no clear indication of how they handle their own translations. We know the Ferengi get translator implants, but since their homeworld is almost exclusively Ferengi one would assume that by the time they leave homeworld they have a working knowledge of Ferengi when they are implanted.
