all throughout colonial history you examples of colonists trading and mixing with locals, for instance in japan, china, the islands around australia, and america.
However having living abroad and experiencing an actual language barrier i've often wondered how the colonists did it without pre existing translations.
This answer is principally for North America.
This may surprise you, but most often, particularly as time went on, it was via translators. Some particularly notable examples like Geronimo de Aguilar or Sacagaw, or my particular favourite, Samoset, exist, but in general trade developed naturally over time. Even before large scale colonization and trade, European fishermen visited (and were shipwrecked) along the cost of North America. Once you have one translator who can speak English or French, finding another who could speak both that person's language and another tribes would be trivial given that trade between Native Americans was extensive. Then you also have men like Étienne Brûlé who spent a great deal of time and effort to learn the languages themselves proactively. Often the explorers and their parties lived in very close proximity with native tribes (particularly if they wintered in North America) and some degree of mutual ineligibility would only have been natural over that prolonged period.
In general, learning the languages operated much in the same way many people learn languages today, not by formal instruction but through consistent and prolonged exposure.
hi! there's always room for more input and specific examples, but meanwhile, check out this section of the FAQ* for previous discussions. There's a subsection "How did colonialists talk with indigenous people?" that's right on topic, the the entire section has useful information:
Cross-cultural communication and lingua francas
*see the link on the sidebar or the wiki tab