That is an interesting question that I hope someone can answer over the full period of time you asked for.

I can tell you that in the very beginning society did not think much about explorers. In fact, the most famous explorer of them all died without fully understanding the impact his discoveries were to have on the rest of history.

After Columbus returned from his first voyage to the New World, the news of his discoveries seems to have mostly interested scholars and officials. King Ferdinand and Queen Isabella were probably very pleased to have some kind of meaningful gains from the voyage, but any kind of average person in Europe would likely not know or not care about Columbus's voyage. We think so because there are only a handful of pamphlets of the news of Columbus's discoveries still in existence (although it's clear they were translated into several languages), which means they were likely studied by the courts and the scholars only. Columbus's discoveries were certainly known to many Spanish navigators and the most privileged among the Spanish government and he certainly did not die in complete obscurity. But little attention was paid to his subsequent voyages as well; almost nothing was published about the rest of Columbus's travels until after his death (1506). It appears that only in 1511 was his story published in all of Spain and began to be paid attention by the general public. From there it spread to the rest of Europe and throughout the rest of the 16th century plenty of works have been written about him.

And there is good reason to avoid idealizing an explorer like Columbus. From his journals, he seems to have been obsessed with gold and would stop at almost nothing to fulfill his ambitions of gaining glory for his name. He was not against lying for dramatics if it meant furthering his own cause. Overall, he was probably not entirely that pleasant of a man to interact with.

We know that Columbus certainly became an important figure in Italy, England, Spain, and the United States during the 17th and 18th centuries, and that by that time people all across Europe knew of his discoveries.

But I am also curious to know about other explorers and their discoveries, and whether most people even cared at all. My guess is that they were also respected, given the sheer number of countries, cities, and places named after explorers.

The Conquest of Paradise by Kirkpatrick Sale is a great book to read for a detailed description of Columbus and his legacy.