What happened between the death of Colombus and the Pilgrims arriving to New England?

A lot. Like, a whole lot.

Who decided to travel west again to reach today’s mainland?

The Spaniards, French, Dutch, English, Swedes, and Portuguese. The Russians came too, but they went East instead of West and did so a long time after Columbus.

[Colombus] never actually touched down on any modern day American land.

As a point of clarity, yes he did. He never touched mainland America but decimated the Taino people of modern Puerto Rico, an American Territory since the end of the Spanish American War over 100 years ago.

There is absolutely no way to cover those 100 years fully in a reddit post, but I'll take a stab at defining some major events... So let's cover some colonial history. Columbus lands in 1492, then makes some return visits. By 1500 multiple nations had sent explorers, the Portuguese sending the Italian Amerigo Verspucci to South America, which he later called the New World coining that phrase, the English sent John Cabot (another Italian) to the North American East coast, being the first European there since the vikings to go there, the Portuguese sent numerous expeditions of folks like Pedro Álvares Cabral who explored the South American coast, as did the Spanish, but they instead primarily worked on developing their Carribean holdings at first. The Pope had settled a debate raging between the two Iberian powers of Spain and Portugal about who had the right to possess the unknown lands populated by heathens and spread the word of God to them, saving their souls from eternal damnation and converting the lands to their potential, as directed to by the scripture (Columbus actually hoped to fund a holy crusade that would herald the thousand year reign of Christ predicted in the Bible from his West Indies profits). The Pope drew a [big line] (https://lh4.googleusercontent.com/proxy/3fRFaT5XUqXFaOdsFTN9gwm5eGuMgNLHbIePtqPK_G4LvhhiDzszRDKaH-yvk3WUhtDc5b60IqXs0CBp5Iwdyg) in the Atlantic and gave the West to Spain, the East to Portugal. In 1494, a year after the Papal Bull was issued, the Treaty of Tordesillas was signed by the two powers slightly moving the line (and this line is why Brazil was Portuguese while the rest of South America was Spanish, it was on the Portuguese side of the line). Then, in Spain in 1506 and likely from heart failure, Columbus died. He still thought he had found the "West Indies" and never learned of what was actually stumbled upon.

Soon the Spanish figured out just how much they had and started their gold hording. Seeing how much they loved gold, some natives used that to their advantage, reporting cities of gold where there were none. Explorers also greatly embellished their findings which led to more rumors. In the early 16th century Spaniards were marching through the North American mainland, Ponce De Leon landing in 1513 on land he named La Florida and [explored] (https://cdn.britannica.com/85/185685-050-EFFBD030/travels-Ponce-de-Leon-1513.jpg) the southern tip and up the eastern coast.

Hernan Cortes took Mexico with the fall of Tenoctilan in 1521, creating New Spain amd unlocking the American West to expeditions. Not long after, in 1527, Cabeza De Vaca (and others) set out on the [Navaraez expedition] (https://www.enchantedlearning.com/explorers/gifs/Devacamap.GIF) that went horribly wrong for the Spaniards, only four people surviving it. At one point they were captured by natives and asked to perform magic rituals for the newly sick natives, sick with European diseases, of course, and would pray for them and recite some latin. Most somehow got better and soon the men were traded from chief to chief as a valuable commodity, eventually being led back to freedom in Mexico by an escort of native warriors. Despite their generosity to the men the Spaniards intended to slaughter or enslave the natives, but ultimately did not after protesting of the idea by Vaca himself - a conquistador that had worked to do the same as his friends proposed to local populations elsewhere in the world. His eight year trek through the southwest had changed him and he saw things a little differently.

Rumors of gold persisted but they hadn't really found any in N America, so more expeditions happened to find the rumored cities built on it. Backing up just a second, you may be wondering how they paid for all of these expeditions. Well, that's where the brutality, enslavement, and pursuit of gold come in - nobody paid these guys to go. They were given Royal charters to pursue wealth at their own expense with private armies, and had to split profits with the crown. No theft, no pay. That's what conquestadors were; land pirates. I don't say that to perpetuate the "Black Spanish Myth" but rather to explain how brutal everyone was, with them having the most native exposure. Anyway, more expeditions: in 1540 Francisco Vázquez de Coronado went searching riches in the [southwest] (https://www.psi.edu/sites/default/files/images/staff/hartmann/coronado/slides/map.jpg), making it all the way to modern Kansas. You get the picture, basically the first quarter (plus some) of the 1500s was a lot of Spanish walking around America looking for gold that just wasn't ever there.

In 1539 Hernando De Soto arrived in Florida with colonists and with them came the first cattle imported to future America (excepting Puerto Rico). They found a man, Juan Ortiz, that had survived the Navaraez expedition a decade earlier and had lived among natives the whole time, but eagerly joined De Soto's forces. They failed to actually establish any colonies but went for a [ramble] (https://exploration.marinersmuseum.org/wp-content/uploads/sites/7/2013/11/De-Sotos-Expedition.png) that would take about 5 years to complete - and found no gold. De Soto died not long after crossing the Mississippi in 1542. A couple more expeditions would happen in the next two decades basically confirming what they had learned; the gold was further south.

Treasure began to pour from the southern continent and pirates - mainly hired by foreign nations - began to notice. By the early 1560s the Spanish were focused on them and created the famed Treasure Fleets, large flotillas with navy escort ships and massive galleons (merchant warships) transporting and protecting the cargo. In June 1564 French settlers made a home near the St Mary's River in La Florida, on Spanish owned land. The Treasure Fleet went from the [Caribbean up past the Florida Keys] (https://www.knowitall.org/sites/default/files/SC-B8low.jpg), right past Ft Caroline - the French settlement - and out into the Atlantic on the Gulf Stream. A Spanish force was sent to destroy Ft Caroline and did so, then in Sept 1565 established the first permanent colony; St Augustine. A similar event had played out further north and not much earlier with the French establishing Charlesfort in modern South Carolina, then abandoning it in 1562 at which point the Spanish moved in and founded Santa Elena where it used to be four years later. The Spanish also traveled into Chesapeake bay at this time, attempting a small mission and settlement, but natives didn't like it and pretty much everyone died.

About this time British ships were probing the Carribean and raiding Spanish ships, led by folks like Sir Francis Drake. Soon a guy named Sir Humphrey Gilbert had the idea to settle lands to the North, securing the fabled Northwest Passage, and meanwhile securing a base to launch attacks on Spanish shipping. He tried twice but it didn't work and the second time he presumably drowned at sea when his ship was lost in a storm. His half brother was a guy named Sir Walter Raleigh and he inherited the Queen's grant to settle lands south of Newfoundland while Humphreys biological brother recieved rights north of there. Both had been inspired by the publications of Humphrey on the purpose to colonize, who had in turn been inspired by Drake - well, partly him... and also the tons of Spanish gold coming across the Atlantic. Raleigh set out to explore and landed on the coast of modern North Carolina in 1584, naming it for the Virgin Queen, Virginia, the first use of that word. They made friends with two natives, Wanchese and Manteo, and took them back to England to learn the language. This allowed future explorers a chance to communicate with other Algonquin tribes, though those two were not the first natives taken to Europe for that purpose (but were really important in Anglo terms). Returning with colonists they started the first Roanoke Colony which was shortly abandoned. They also attacked St Augustine quite sucessfully leading to a Spanish withdrawl back to Florida, abandoning their South Carolina colony to reinforce St Augustine and hold all mainland grounds south of there. Raleigh would return again in 1587 and establish a second colony, this being the lost colony of Roanoke and the first English attempt at a permanent colony in the future US.

There is certainly much, much more to it since as you can see there were many nations moving in many directions with different motives. But as far as major European events in future America in the 1500s, that's the highlight reel. Of course Jamestown and the failed northern partner colony would come in 1607, then the Dutch landed in modern New York, the French expanded in Canada, and the Swedes began to eye spots as well. In 1620 the Pilgrams set sail.

Let me know if you want more depth on anything in particular, and a good overall view is American Colonies by Ann Taylor.