Specifically, would there be any qualifications or disqualifications to joining a conquistador expedition? For example, would they take any European recruits who wanted to join or only subjects of the Crown of Spain?
Also what could I expect to be my reward? Land? Gold? Would knights or other nobility join expecting larger rewards than an average soldier?
So you want to be a conquistador? Excellent. There are a few things that you need to think about before you try to ship yourself off to the Americas.
First, where are you from? Is it within the Spanish realm? Depending on when in the 16th century you exist this could be parts of Italy, Portugal, Dutch territory, parts of the Holy Roman Empire, and islands in the Mediterranean. Does it matter? Not really. You could be Greek, French, Polish, or Irish for all the Crown cares. The key here is to answer a more serious question: Are you a Christian? This is a loaded question, of course. To begin with, there is a difference between Old and New Christians. If your family is from Iberia this question is particularly relevant to you. If you are from France, Italy, or the Germanies, there is a different meaning behind the question. Are you loyal to the Pope and the Catholic Church? Do you support the radicalism of Luther and his followers? Should you abandon the "Sacred Faith" and protest alongside the growing number of radicals in the Holy Roman Empire, you should just stay home for now. For the moment, let us assume that you are from Extremadura in the Kingdom of Castile. Your family, having fought in the wars against the Moors for generations, is comfortable in knowing that they are Old Christians and safe from the Crown/Church push to investigate the newly converted Jewish and Muslim residents of the peninsula. Let us also assume that your father is a man of local importance, but not too powerful politically. Perhaps he is a celebrated artisan or a successful merchant. He has contacts in Seville, Madrid, and Lisbon, but he is not elite and never will be. For our purposes, let us assume that he has excellent contacts with some of the major German or Italian merchant houses in Europe. The Welsers or Fuggers in Bavaria or perhaps the remnants of the Medici firm. With these contacts, you are able to get your foot in the door when it comes to small favors.
Another question must be asked as well: Why do you want to leave? Before 1519 there are rumors of great cities and riches to be found, but El Dorado-type stories are only fantasies. Life in the New World is difficult. Keeping indigenous peoples working in the fields and mines around Santo Domingo doesn't sound too promising a career and fighting indigenous warriors on the islands of Cuba and Puerto Rico is dangerous. Perhaps by the mid 1520s the riches of Mexico have attracted you. The great land masses west of the Caribbean are relatively unexplored. Florida, Venezuela, Santa Marta, and the Yucatan need conquering and pacification. It will be difficult. Why would you leave the prospect of carrying on your father's legacy and staying somewhat safe in Spain? Do you have older brothers? Is your father's business being bought out over time? These are important issues. Again, let us assume you have a few elder siblings and they have not only placed their claim on your father's business, but one has also used dear padre's connections to secure a position in the clergy. Your prospects in Spain are not good.
So what do you bring the Conquistador table? Your father believed in education for his sons. You will not attend a university at age 14/15 like other first/second sons of means, but you have the ability to read and write. You pick up a little Latin as well from hanging around your brother the priest and your older brother the future merchant. You pick up some basic law education and maybe have access to some classical works like Plutarch, Cicero, and ancient histories. To put it simply, you are educated. The New World will need educated men. You have the potential to make a name for yourself with that kind of background. What else do you bring? With your connections you have a couple of horses at the ready to take with you. You manage to get some nice boots, a good knife, and a reliable sword as well. Where is the port?
Not so fast. You have to go to Seville first. If you were living in the Holy Roman Empire you would first have to travel to Antwerp and take a few different ships to get to Iberia. Thankfully, Extremadura is fairly close to Andalusia and Seville. In Seville you will have to go to the House of Trade, the central hub of any and all business dealing with the Americas. It is a stone's throw from the Cathedral and the Royal Palace so it gets pretty busy there. You wait your turn to speak to an official who will decide if you get a license to go to the New World. You cannot just go. Things must be orderly, done in duplicate or even triplicate, and often money must change hands in various modes of legality. You are lucky, however. Your family's reputation is known by a few local officials and character references are easy to obtain in order to prove your devotion to Holy Mother Church. You are registered in the royal record books for all time as LateNightPhilosopher, son of DaytimePhilosopher, resident of X, sailed for Santo Domingo in the ship El Glorioso Reddit. This may be the only piece of paper proving your existence. (The Spanish archival system is superior to most other nations, thank God.)
You arrive in Santo Domingo after making a stop in the Canary Islands. You have had a chance to get to know the other men on the ship. For the most part, you are all either from Extremadura or Andalusia. There are some Catalans and Basque on board ship, but you outnumber them. There are quite a few Flemish and Germans as well. You are educated, but you cannot quite get the Germanic words to come out of your mouth. Hans Schiller, a man who comes from a similar background as yourself is perfectly willing to go by the name of Juan Aleman if it makes it easier for you. Hans and various members of the ship disembark at Santo Domingo, but the ship continues with you to Cartagena. You register with the town council and immediately find yourself a vecino or resident of the city. Your horses and education make you quite popular. If you sell the horses you will have plenty of cash to invest in some conquest expeditions or to simply start your life in a comfortable house on the coast. With your education you could be a notary, a merchant, or a town official if you played your cards correctly.
You find yourself introduced to an old timer. One of the early conquistadors of Cartagena and Santa Marta who survived the Rodrigo de Bastidas expeditions of the 1520s. He invites you to join his entrada or expedition. Will you get paid? No. There is no money for a salary. However, if you help in establishing a pool of money, you will receive a cut of the spoils of the expedition proportional to your investment. With a horse? You will get a larger share since the animal requires extra care. Thinking it over, you sell your weaker horse. You get some ready money and keep half of it as a backup in case you come back with nothing. The other half you put in the pool. You still get the prestige of being a horseman, but now you have put cash into the pot. You are young, maybe 18 or 19, but you have invested nearly everything you have into a "company" system. Nearly all conquest endeavors were set up in these systems. The Crown was not putting up money, only allowing certain parties to direct and operate territory in the name of the Crown. The individual conquistadors had to invest their own cash or simply join a company with only a sword and some boots to their name. Their cut would be dramatically less than yours, but if the entrada finds a city of gold...well you will both be set for life.