This is an interesting subject. The Jesuits did a lot of the early missionary work, and their story is well attested. Off the top of my head, (with regard to the Jesuits):

1. They were criticized for downplaying the transsubstantiation aspects of the Eucharist, probably because they didn't want any backsliding into ritual cannibalism.

2. They came to North American expecting to find the Devil's agents everywhere, and ended up finding just people - no more good or evil than any other peoples. This helped guide the Jesuit worldview from the medieval outlook towards the modern outlook.

Aha, the spiritual conquest. This is something that was extremely varied and is still widely debated today. However, I'll answer as best as I can.

The first priests in the New World were Franciscan friars who essentially performed mass baptisms without even the bare bones of a catechumen. All that was required was anointing with holy oil, a recitation of baptismal liturgy and the physical act with the water, and the Franciscans at times did this in a form of conveyor belt system, a constant stream of Indians being baptised. However, although it was rapid and stripped down, no pagan entered the Church without some form of preliminary training in the essentials: a single omnipotent God, eternal, all-good and all-knowing, Creator of everything; the Holy Virgin; the immortality of the soul; and demons and their perfidies. The Dominican order complained about this rapid form of baptism and demanded that the indigenous people receive more education in the tenets of faith before being baptised, and eventually a middle ground was struck.

The central tenets were not so much changed as stripped down to the essentials. The issue wasn't about getting the indigenous people interested, but getting them to understand it. This is why the priests led the way in learning, translating and recording grammar in Nahuatl, so that liturgy could be given without an interpreter. The Franciscans wanted to get them baptised, Dominicans wanted them to understand the religion a bit more before this, and both wanted to leave no room for the survival of native religion. While the missionaries had originally toyed with syncretist notions, such as comparing the figure of Jesus Christ to Quetzalcoatl, this could and did lead to continued idolatry (as well as toed the line between metaphor and heresy) and the practice was quickly dropped.

The form of Christianity that was taught depended on geography as much as policy. In Mexico City and the surrounding large urban areas there were large churches, plenty of priests and you could at least have some form of control over indigenous faith. Out in rural areas where it was easy to travel, people tended to see a priest once or twice in their lives - a community would be baptised and a 'check-up' performed a few years later to make sure they were still Christian. Naturally, these places saw a great deal of continued idolatry and use of indigenous shamans and rituals. However, in rural areas where it was hard to travel, such as the Andes, it was quite the opposite. Those who converted actually had a great deal of instruction in the tenets of the faith because typically the village/community would have an itinerant friar.

To return to your question, no major changes were particularly made to adapt the Christian faith for the New World, and this is generally seen as a fundamental flaw in the missionary efforts. The conviction that the indigenous religions had been spawned by the Devil led the priests to believe that nothing could be salvaged for Christianity from them. This, combined with the simplification and stripping away of doctrine, confined Christianity to a theological box impermeable to indigenous culture, inseparable from its European cultural, political and social framework, and it is argued that this is what allowed indigenous religious beliefs and idolatry to continue for such a long time after the conquest.

Hope I've helped or interested you in at least some way, and here's some suggested reading for the period immediately after the conquest:

Robert Ricard, The Spiritual Conquest: an essay on the apostolate and the evangelising methods of the mendicant orders in New Spain, 1523-1572 esp. chapter 4, 'Prebaptismal Instruction and Administration of Baptism.'

Sabine MacCormack, '"The Heart Has its Reasons": Predicaments of Missionary Christianity in Early Colonial Peru', Hispanic American Historical Review 65.3 (1985), pp.443-66.

J Jorge Klor de Alva, 'Spiritual conflict and accomodation in New Spain: Toward a typology of Aztec Responses to Christianity', The Aztec and Inca States, 1400-1800 (New York, 1982), pp.345-66.