If I was going to arrive in a foreign country, I’d try to gather as much information as I could, especially if I anticipated I’d be visiting a royal court as an ambassador. Was there any valuable information about Japan and China at this time that they could have brought along?
I don’t have a specific answer about Columbus, but in general, European explorers in India didn’t put a lot of effort into establishing good relationships with the people they encountered. It’s not unreasonable to expect Europeans to know more about the Indian Ocean/Asian world they were entering, given the long history of trading links between them, but they don’t.
A few useful pieces of context: First, Europe is still emerging from a few centuries of relative isolation, and this is more extreme for western European countries like France, Britain and the Netherlands who are geographically on the edge of Eurasia. (Spain's a weird case, since they've been part of an Islamic Empire, but have also just forcefully expelled it so don't have friendly relationships with former north African and middle eastern connections.) So Europeans know Asia exists and has luxury goods, but they don’t necessarily have the resources like dictionaries or histories to learn more about it. There is trade passing through the middle east to western Europe (primarily via Italy and the Ottomans), which means some people had the linguistic and cultural knowledge to interact with Asian courts, but those are going to be minority enclaves. Second is the fact that most of these expeditions are not official state visits or diplomatic encounters, they’re trading missions sponsored by states. So even if they did have the information, cultural competence wasn’t necessarily a high priority for them-- they just want to communicate enough to trade. In terms of gifts, Europe doesn’t actually have a lot to offer Asian courts at this point; the whole reason they want to get to Asia is because they have luxury goods Europe doesn’t. There are even some examples of Asian courts being kind of insulted by the gifts European visitors do bring in the 1500s.
This wasn’t part of the question precisely, but another group on the ground in Asia are Christian missionaries. You might expect them to be more motivated to learn about the populations they’re going to try to convert and learn languages-- but many of them are still very bad at it. A great example of this is the case of Jesuit missionaries in the Mughal court, described in the Alam and Subrahmanyam article listed below. Basically, there’s a Jesuit missionary in the court of Mughal Emperor Jahangir who keeps writing back to his superiors saying he’s on the cusp of converting him; but if you look more closely at what questions the Emperor is asking him, it seems like the Emperor already knows a lot about Catholicism, has no intention of converting at all, and is actually pointing out theological inconsistencies in the missionary’s understanding. This is part of a large pattern of Mughal emperors essentially debating Jesuits for entertainment, which Jesuits seem to not pick up on. Again, this might partly be the product of lack of information in Europe about Asian courtly conventions; it might also partly be the product of lack of interest in learning more about them. If anything, some Asian courts learn more about European visitors than the reverse.
Source: Alam and Subrahmanyam “Frank disputations: Catholics and Muslims in the court of Jahangir (1608–11)”
Regarding language, most of the expeditions made during the Discovery era had at least one interpreter, that knew at least Arabic, but often this didn’t work out, as the sailors found places which hadn’t been contacted before. In one of the first voyages the Portuguese made to Africa they brought with them a Moor that had converted to Christianity as an interpreter, but his usefulness diminished as they went south. He knew Berber, but the Berber dialects of what is today West Sahara were already very different from what was spoken in northern Morocco. In the end, the Portuguese captured some of the people they found as slaves, and some of these slaves were meant to become interpreters. They were taught Portuguese and would accompany the Portuguese in future voyages. When the Portuguese reached the eastern part of Africa, the usefulness of Arabic speakers again increased, and they used these interpreters again in their dealings with Eastern African kingdoms, Muslim states and Indian states.*
Columbus did something similar. His interpreter in his first voyage was Luis de Torres, born Yosef Ben Ha Levy Haidri. He was a converso, he was born Jewish but converted to Christianity in 1492, the year Spanish Jews were given the choice of being expelled or accepting Christ. He spoke many languages, among them Portuguese, Spanish, Arabic, Mozarabic and Hebrew. When his efforts didn’t bear fruit, sign language was used by the sailors in Columbus voyage. Later Columbus would resort to the Portuguese technique of capturing or enlisting natives and teaching them Spanish to make them the future interpreters. One of these would get the name Diego de Colon (adopting the surname of his godfather, Christopher Columbus) and would accompany Columbus in other voyages to the New World (he would translate the dealings with the native chiefs of Hispaniola, nowadays Haiti and Dominican Republic).
The Europeans had no knowledge of Japan at all, beyond what snippets Marco Polo had written about it. No European set foot in Japan before the arrival of the Portuguese in the 1540s. Many Europeans had arrived in China before that, but the land route to China had been sparse for some time due to the collapse of the Mongol empire. The sea silk route was much more traversed, and was used by Muslims speaking Arabic. The Portuguese Pero de Corvilhã was given the mission of making the land route to India in 1487 and was successful, sending a letter to the Portuguese king in 1491, but he didn’t arrive on China or Japan, so knew nothing about that. He learned a lot about the merchant routes of India, though, so when the Portuguese arrived by boat in the Vasco da Gama expedition they knew that Calicut was the biggest trading city and they should try it first.
Regarding presents, as other users have mentioned, the Europeans didn’t have that many gifts at first that impressed the Asian rulers. In the 1600s European clocks would be a marvel in China, but the early 1500s in Asia their gifts were underwhelming. Vasco da Gama arrived in Calicut in 1499 and was received well at first by the samorim (the ruler of the place). But the samorim was disappointed by the gifts he brought, namely: 12 pieces of cotton cloth, 4 scarlet caps, 6 hats, 4 coral necklaces, 6 bathtubs, 1 chest of sugar, 2 barrels of olive oil and 2 barrels of honey. When the samorim said he was disappointed with the gifts and didn’t believe the king of Portugal was rich, Vasco retorted that he was a diplomat, not a merchant, and would come back later with better gifts. Vasco was allowed to buy some spices in Calicut, and even this small quantity gave the expedition a very high profit. Later the Portuguese would return with more gifts, but what they had that impressed most the samorim later would be their ships (the naus). Vasco would return with many more (the first expedition had 3 ships, by the 1520s Portugal maintained something like 20-30 ships in the Indian ocean all the time) and would spread terror among many rulers of the Indian ocean, including the samorim of Calicut, who became part of an anti-Portuguese alliance.
Columbus in his third voyage to the New World actually expected to arrive in Calicut as well and to meet Vasco da Gama there. He went south, thinking he would find a route to India, but only found Venezuela and many other islands and places in the Caribbean ocean. The second voyage was the one that was bigger (had 11 ships), the third voyage was smaller (with 6 ships that became 3 ships for Columbus expedition and 3 to replenish Hispaniola) and probably didn’t have better gifts than the Vasco da Gama expedition had.
Regarding customs, they knew basically nothing. The Vasco da Gama expedition when they first met Indians in the eastern coast of Africa thought they were Christians because they were saying their god was named Krishna (which they though meant Christ). They thought the gods depicted with many arms in the temples in Calicut were saints. The Portuguese also bowed and prayed to a Hindu goddess thinking it was a depiction of the Virgin Mary (some think it was Mariamma, goddess of death and who gives protection against diseases, and the similarities in names confused them). On the other hand, the Indians at first also thought the Christians were some kind of Hinduists.
* By an incredible coincidence the Vasco da Gama expedition of 1498-1499 found two North African arabs, born in Tunis, in the indian city of Calicut, and they spoke Castillian (Spanish). This wouldn't be of use, though, as these Arabs didn't want the Europeans to trade with the Indians, and tried to convince the Indians to expel the Portuguese.
A previous answer from a user who has [deleted] his account responds to this. Basically, Europeans did have guidebooks, especially the travels of Marco Polo.
https://www.reddit.com/r/AskHistorians/comments/ggf06q/did_columbus_really_think_he_was_in_india/
/u/JustePecuchet wrote, a decade ago: