I understand that many languages have many things in common and that the effect of trade, religious spread, and politics/war would motivate people to learn the languages of the peoples that they are dealing with. What I am really curious about is how people actually learned new languages when they had no medium to speak between, no common language, and no languages with large similarities.
Was there more often than not a common language for people to use together, and then use that language to learn each others native language?
u/myfriendscallmethor's language acquisition compilation is useful for us here, but the real thing to consider is u/Iphikrates' reminder further down in the thread. That is, the situation of "they had no medium to speak between, no common language, and no languages with large similarities" is vanishingly rare, and Not How It Works for the majority of inter-language contacts.