Due to the impact of centuries of globalisation even before they met another species, every species in the galaxy has a single language which serves as a lingua franca across the entire species. Their citizens are educated in it from birth, even if many are also fluent in a local dialect. While there are, of course, those who do not understand this language, they are very much in the minority.
For the humans, for example, a single language had largely caught on by the 22nd century and by the 24th was in almost universal rotation. Regional languages still remained and many are still used for practical work today, but the common "Human language", as it is usually called, is dominant.
There is, however, no common language between species. As the different species have such different cultures, a common tongue would be difficult to agree upon. And there are physical problems: what a Twis'rin has no trouble pronouncing would take considerable practice from a Balorim, and of course a Captab can't speak at all.
Therefore, most foreign nationals make it a habit to pick up the native language of whatever species they're doing the most work with. In cases where physical factors make speaking or understanding the language difficult, small translator robots are typically used. Except for very expensive and hard-to-find models, these do not have the ability to actually translate sentences and phrases, but a skilled operator who knows the language he wishes to use can communicate quite effectively with them.