When Columbus first met the Natives, how did they communicate? Was it just frantic chaos?
That's a really good question. One piece of writing comes to mind. It's a Jamestown settler describing first contact with a native group. In this case, it was quite friendly. Communication seemed to be pretty difficult, but they used sign language as best they could to help each other understand one another's language. This settler wrote that the natives "laboured very much by signes to make vs understand their Languadg."
Obviously in modern, standardized spelling that's "...labored very much by signs to make us understand their language." You can see that excerpt here.
But overall, communication was quite difficult. Not only did they not understand each other's language, but they had very little context to understand one another's culture either, so even sign language could be taken completely wrong. It was one of the many reasons why there was so much violence between settlers and natives. So much potential for misunderstanding.
From what I know about the French colonizers in Canada is that the Algonquins were able to pick up the French language very fast. Which then the French learned their language.
The French had early missionaries who would go live with the Algonquins, Hurons, and others down towards the Great Lakes.
The priests made real efforts to learn local language and teach French in turn.
Of course some of the missionaries were nice people, though there was tension because Europeans brought disease. No one understood disease so the locals would unsurprisingly blame the European God/ blame the missionaries. You'd be surprised how often though the locals liked the missionaries and were interested in learning about the rest of the world or teaching the French about North America.
Other missionaries were very patronizing (as the Church is wont to do) and tried to force the Bible onto locals. This usually went poorly when combined with the spreading disease.
There's always room for discussion, but perhaps the section Cross-cultural communication and lingua francas in our FAQ will answer your inquiry.
Not the answer you're looking for, but, like in most cases, a pidgin language is developed but not without a lot of miscommunication. In the portuguese hereditary capitanies [capitanias hereditárias] that would later become the brazilian colony, the portuguese explorers [bandeirantes] borned in brazilian territories would learn the Tupi language that, naturally, became what was know as Língua Geral Paulista, very similar to the Nheengatu (spoke today by 8k people or so), a pidgin (mix) language the incorporates Tupi-Guarani and some Latin and Portuguese. According to Sérgio Buarque de Holanda, Portugal's Secretary of State marquis of Pombal was the responsible for banning the use of the language. But at some point until the 18th century, most of the europeans who were born at the colonies spoke Portuguese and the Língua Geral Paulista to communicate with the indigenous natives, even though Portuguese was the official administrative language. After that, Portuguese became the dominate language and the indigenous started learning it too (with a lot of coercion, specially from the jesuit order). Speaking about the jesuit, before being expelled from Brazil in 1759 by the same marquis of Pombal, they were the first to make a Tupi-Latin dictionary and teach portuguese to the natives.
Sources: Sérgio Buarque de Holanda, Raízes do Brasil (1936)