In the absolute earliest encounters between Europeans and Native Americans (between Columbus and Caribbean tribes, or Pero Vaz de Caminha and Native Brazilians), there was no initial 'dialogue' but rather attempts at rudimentary communication through body language and gesture. For example, when Vaz de Caminha's company arrived in Brazil, they held a mass and a number of Brazilians are recorded as participating, but only because they mimicked the movements of the Portuguese worshippers. Also in this case, and in many others, initial contact was marked with an exchange of goods; for example, the Portuguese gave the Brazilians their hats and a rosary, and in return, the Brazilians gifted them with valuable stones. Actions like these were meant to create the potential for a cordial interaction, or at least to stave off violence.

You might be surprised, however, how early on some people in both Europe and the Americas learned one another's languages. When Cortes landed in 1519, he came with a Spanish interpreter who already knew a certain Maya dialect in order to communicate with the Natives he expected to encounter. Before arriving in Aztec lands, Cortes and his company had stopped in a Mayan area and been gifted with slave women for them to take as concubines. When Cortes and his Spanish interpreter landed on the coast of the Aztec empire, they found themselves totally useless at interpreting and communicating to the Native people, who spoke Nahuatl. Luckily for them, one of their concubines knew Nahuatl and Mayan - so she interpreted for the Natives to Cortes's Spanish interpreter, and he translated those conversations into Spanish for Cortes. Before long, however, the concubine, whom we now refer to as Malintzin or Malinche, learned enough Spanish to become the primary interpreter. She's a controversial figure in Latin America to this day (see Malintzin's Choices - Camilla Townsend).

Other sources I'm using here: Colonial Latin America by Kenneth Mills; Columbus's letter to King Ferdinand: http://xroads.virginia.edu/~hyper/hns/garden/columbus.html

For Columbus, it was actually through Luis de Torres, a Spanish Jew who was skilled in multiple languages. He dealt with the communications and learned new native languages. Also, all expeditions had priests come with them who are already excellent at learning new languages, as reading Latin, Greek, and Aramaic were essential to studying scripture. Priests were mandatory for expeditions to tend to the spiritual needs of the sailors.

Explorers were well aware of the language barriers and thus employed translators (usually veteran sailors, merchants, or royal dignitaries) and clergy to assist them.