There have been a number of studies on early conquistador companies. The most surprising aspect of most participants is that they were generally not trained soldiers. In fact many companies were composed largely of tradesmen and professionals. The famous Herman Cortes was trained as a lawyer.

There was good reason for this. They were not members of any standing army. In fact conquistadors were private citizens who had joined a company that was licensed by but not financed by the king. This meant that conquistadors had to pay their on way. The need to pay for transport, supplies, weapons, etc precluded the participation of unskilled common Spaniards.

I can't speak to the warfare practices of Caribbean groups but certainly in Mexico and Peru, there were trained warriors and imperial armies. But Spaniards had a number of technological advantages. Steel swords were much better weapons on the battlefield than most native hand to hand weapons. Gunpowder weapons were powerful for their shock and awe effects, less significant for their ability to kill or be accurate. Horses were very helpful when battles took place on open terrain.

The other major advantage held by the Spanish was their success in recruiting native allies. Spaniards actually learned native language quickly and used that knowledge to determine local politics. By exploring rivalries and antagonism between neighboring groups they were able to amass more native allies than they had Spanish participants.

Some good reading material:

Restall's Seven Myths of the Spanish Conquest

Lockhart's Men of Cajamarca

Schwaller's First Letter From New Spain

For more on Carib warfare you can look into works by Neil Whitehead including Lords of the Tiger Spirit