In-Universe: The Advent of the Universal Translator mean's you hear everyone speak what ever languge your where rased with, like if your Japanese, you would hear everyone speak Japanese.
Real-World: It easier to just write everything in english. The rest of the languages would still be studied, however, everybody would so greatly overestimate how hard they are to learn that they would become extinct by the 23rd century.
We did hear Data speak French in the Star Trek: The Next Generation episode "Time's Arrow", and Q often addresses Picard as "mon capitaine" ("my captain"). Picard also is heard to say "merde" ("shit") on one occasion (the French word got past the censors in the U.S.) The Universal Translator probably translates them perfectly since there must be complete data on even the colloquialisms and cultural idioms of every Earth language. The Federation might use a standard, contrived common language, such as Esperanto.
When Star Trek is telecast in non-English-speaking countries, it is dubbed in the language of that country. Star Trek Region 1 movie DVDs are known to contain at least Spanish and/or French (the other two North American languages) subtitles.