Perhaps the most significant "meet and greet" in history was the physical exchange of plants, animals and other organisms, plus ideas and technologies, that followed Columbus' "discovery" of the New World in 1492: the so-called Columbian Exchange.
The Spanish, and subsequently other Europeans, brought to the New World horses, domesticated cattle, sheep, goats, pigs and chickens, and then returned home with American animals new to them. The native Americans were particularly awed by the conquistadores' horses and dogs: the almost invincible Spanish cavalry proved to be a major factor in the conquest of the Americas, while the fearsome Spanish fighting mastiffs were a far cry from the little Mexican chihuahua and xoloitzcuintle-type breeds of domestic dog which were mostly raised for food. But while American turkeys and guinea pigs might be novel food animals, and parrots a talkative source of colourful feathers, the llamas and alpacas didn't much impress Europeans who already had oxen, donkeys, horses and mules as beasts of burden, and millions of sheep as prolific wool providers. Probably more significant for both sides in the longer term were the exchanges of food plants. Wheat, rice, onions, sugar cane, bananas, citrus fruits, mangoes and coffee would revolutionise the agricultural economies of the Americas; while potatoes, tomatoes, chilies, maize, cocoa and vanilla would transform the cuisines of the Old World.
Not all newcomers, to whichever side, were so welcome. Although its New World origin is still disputed, there is not much evidence for syphilis in the Old World before Columbus arrived back from his first visit to the Caribbean. Within just a couple of years of his arrival back in Seville (in March 1493) there was an outbreak amongst the Spanish soldiers defending Naples against the besieging French army of Charles VIII (the first unambiguous descriptions of this 'new' disease seem to be those as occurring in late 1494 amongst mercenaries then newly arrived from Spain). This new and unusual disease soon spread to the besieging French army, and from there into the rest of Europe when the French army eventually retreated from Italy a couple of years later. The French called it the Neapolitan disease, the Italians, Germans and English called it the French disease, the Dutch called it the Spanish disease, while the Turks called it the Christian disease. As it spread further afield into the Middle East and India, the Muslims blamed the Hindus, and visa versa. But ultimately it seems to have come from Central/South America and been carried to Europe by the return of Columbus' very first expedition to the Caribbean.
Syphilis, nasty though it is, is a very slow killer: far more devastating were the diseases America received in a very unfair exchange. The first Spanish conquistadores brought with them all the common illnesses that had long plagued the Old World and to which they themselves had acquired some immunity: measles, chickenpox, smallpox, bubonic plague, typhus, malaria, scarlet fever, diphtheria, influenza and the common cold - later to be joined by yellow fever and dengue, carried across the ocean by a combination of West African slaves and West African mosquitoes. Within a decade of their first contact with Europeans - and often very much sooner - native American populations were suffering epidemics of many of these Old World diseases to which they had little resistance. Accordingly the indigenous populations often fell by over 90% within just a few years of their first contact with the invaders. Indeed direct contact wasn't even always necessary and these new diseases often worked as a microbial vanguard advancing ahead of the conquistadores. Smallpox for example had ravaged the Inca Empire two years before Pizarro first arrived off the coast of Peru. When he landed he found the mighty empire to be already weakened, both by the epidemic itself and by the civil war that had followed the sudden death by the disease of both the king and his principal heir, leaving two younger sons to slog it out for control.
Back and forth the exchanges between the Old and New Worlds continued over the centuries, sometimes imparting benefits and at other times wreaking havoc, often subtly, to one or both sides.
For example the Spanish had arrived in the Americas searching for gold and silver, which they found in quantities that far surpassed their initial hopes. This provided a financial boost to great building projects both in Spain and elsewhere, but it also financed over a century of needless religious warfare. Also, while the exploitation of these riches meant brutal enslavement for the indigenous people of south and central America, the flood of precious metal into Spain created such inflation that many ordinary Spanish lost their livelihoods and effectively become little better than slaves themselves. Furthermore Spanish agriculture and industrial production were so undermined by this inflation that it would be centuries before they recovered.
Potatoes from the Andes and maize from central America certainly transformed the diets of some of the poorest people in the Old World, but they brought with them their own problems. A reliance on maize combined with an ignorance on how to process the crop to get its full dietary benefit (nixtamalization) lead to multiple outbreaks of severe vitamin deficiencies (manifesting as the disease pellagra) in areas of southern France, Spain, Italy and North Africa. Meanwhile an over-dependence on potatoes in part created the conditions for the devastating Irish famine, when potato blight (which itself had quietly travelled to Europe along with potatoes) caused widespread crop failures. Finally the same New World forests that produced quinine - a vital medicine for treating the Old World's malaria - also held the coca and tobacco plants that eventually gave the world cigarettes and cocaine, with all the social problems that are still associated with them.