I'm sure somebody can speak more about the Spanish policies and actions in their newly subjugated regions, but I would make a quick note of the dynamic that had developed early on between the Spanish and natives: wherever the spanish went, so did disease. Particularly in the Aztec territories (as the disease started spreading faster than Europeans did, beating them into North and South America), there was a lot of religious turmoil surrounding the Spanish. While the theory for a long time has been that the disease was so effective because the native populations had built up no resistance, the new theory is that they simply didn't know how to respond to a disease at all. In the limited Aztec records left, we have extensive records of significant calamities, and not a single mention of any sort of plague or epidemic. The Aztecs did not know what a disease was or how it operated, especially one as lethal as smallpox, and it meant that they never changed their living structures to accommodate it: they would sleep with the infected in communal huts, trade with them, handle their goods, eat with them, have sex with them. They treated them as they would any other member of society. Suddenly your entire city is infected, people are dying in agony en masse, and you don't know why. Meanwhile, you're also dealing with a newly arrived peoples who arrived on ships so large their essentially houses on water, look different than you've ever seen, and wield weapons and armor you can't understand. They are also crucially not affected by this new disease/curse. It was certainly a common connection that these men were either divine, or messengers of the divine, and that the fact that they were immune to the curse that was killing your city was seen as evidence that the Gods were unhappy.

It's certainly a sentiment that was seen in Tenochtitlan before the city fell to Cortes, and further seen as Cortez took advantage of it to diplomatically win over/conquer neighboring cities and territories. There was a sense of religious fear among the natives that these new people were divine, and that the Gods were unhappy, and that led to an increased conversion rate because as far as the natives were concerned, they had already seen evidence of the validity of these beliefs: they had protected the spaniards, and slain the other natives around them. European missionaries would note how docile the natives of this region had become in religious matters, and how susceptible they were to conversion.

But I can only speak in that one specific regard about the Aztecs. I can't speak for other native civilizations, nor for what I'm sure were rather extensive Spanish efforts to convert and destroy native culture.

Can't speak for other areas of Spanish rule, but I would argue that in California they weren't very successful at all. In California, forced conversion and slavery were part of the Mission system of colonization. Natives sometimes revolted, and used violence as a means of resistance. In addition, they used other means to resist for example Natives frequently ran away and frustrated church fathers by simply ignoring the strict Catholic sexual morals and prohibitions on relationships and cohabiting of the genders even at the risk of flogging.

The Spanish used extreme violence and virtual slavery to control Native California populations and attempt to force them into a so-called 'civilized' way of life. The fact that such terrible violence was used leads me to believe that Natives weren't so willing to accept Christianity as you suppose, and only the threat of torture and death was enough to convince them to convert. That they continued often to follow their own codes of conduct, though (as I mentioned they often followed their own codes of sexual conduct) makes one doubt how serious many of these conversions were.

I think your impression that natives quickly embraced Christianity is simply a product of the 'history is written by the victors' effect, as California natives weren't literate and unable to write down their own views about the Spanish.

Please read this article for more information and sources.

http://www.sandiegohistory.org/journal/78spring/labor.htm

EDIT: Bad writing