Communication during initial contact was generally limited to basic signs and a lot of pointing at things and repeating its name, which doesn't make for great conversation. So involuntary immersion was usually the way people crossed this huge language barrier. Columbus captured eight Taino on his first voyage in order to force them to learn Spanish and serve as interpreters on future expeditions; only one actually returned to the Caribbean. Cortes' first significant interpreter was a Spanish priest that had been shipwrecked on the Mexican mainland years before. Once some initial interpret is established, it becomes easier for others to learn the various languages as well.
In this post I go into more detail of how the language barrier was crossed in the specific case of the de Soto entrada into the American Southeast, circa 1540. There are three distinct phases of interpreters used in that expedition: captured interpreters who were forced to learn a small amount of Spanish prior to the expedition beginning, a shipwrecked Spanish sailor who had lived in Florida for years, and a young Native trader who already spoke several languages from the region and was able to add Spanish to the list with the help of the aforementioned sailor.