Needless to say all the advantages we now have to learn a new language in the 21st Century, my question is mainly focused on how did explorers from the past (whether it be really far away or not-so far away from our current timeline) learn new languages (from foreign/close cultures). Thanks!
You may get more answers if you posted in /r/AskAnthropology or /r/Linguistics. From what I remember from my undergrad linguistics class, a pidgin forms between the two - a simplified way of communicating (points to apple, says "apple," other person understands what you are talking about), but it's not a language. From there, I can't remember much, but you should have luck at either of those subs.
From the voyage of Ferdinand Magellan, we have a great account written by Antonio Pigafetta who was a supernumerary and chronicler of the expedition. In it, he details how he would work out a vocabulary for heretofore unknown languages. It's remarkable as there is no precedent for what he was doing, Anthropology wasn't yet a professional field of study at the time.
It's mentioned in Lawrence Bergreen's "Over the Edge of the World: Magellan's Terrifying Circumnavigation of the Globe" Upon encountering a fabled Patagonian Giant and taking him aboard ship, Pigafetta worked with the captive to establish a working vocabulary and pronunciation.
From page 209: '"When he, asking me for capac, this is to say, bread (for so they call the root which they use for bread) and oli, that is to say water, saw me write down these names, and afterward, when I asked him for others, pen in hand, he understood me." Their collaboration resulted in a phrase book called, "Words of Patagonian Giants."'
Pigafetta also documented some phrases in the Cebuan language of the Philippines, complete with numbers which was critical as the expedition sought to trade and establish new trading relationships. I like the authors phrasing here...
From page 267: "As before, Pigafetta worked in a vacuum, guided by the dictates of his intuition and common sense, since there was very little precedent and absolutely no professional standard for the ambitious task of writing down words and definitions for an entire oral tradition. Despite these obstacles, he managed to devise a phrase book that might be useful for subsequent expeditions that happened to pass through Cebu."