How did they draw native attention? How did they teach? How were their locations chosen? Did their target audience change over time as new groups of people arrived?
I am not as familiar with the mission towns of Mexico, but I can talk a little about the missions along the northern frontier of the Spanish Empire in modern-day Florida and New Mexico.
After the excesses of cruelty seen in the initial years of contact, the 1573 Comprehensive Royal Orders for New Discoveries, placed missionaries at the forefront of exploration and pacification of new lands. Instead of valiant conquistadores rampaging into new territory, the Orders prohibited the entry of unlicensed entradas into new lands, and prohibited violence against Indians, under threat of fine or death. Entradas were to be led by conquistadores of the spirit, most often by the Franciscans along the northern frontier of the Empire, though Jesuits briefly established missions in Florida, as well as longer-lived missions among the Pima in southern Arizona.
The frontier missions obvious purpose was to provide a spiritual harvest for the Catholic faith, supported financially by the Spanish Crown. A secondary role was to provide an economically expensive, but vital, frontier presence against encroachment from other European nations, and protect interests further south. New Mexico and Texas provided a supportive buffer for the lucrative mining enterprises in Northern Mexico, and Florida provided a safe haven and support for ships crossing the Atlantic. Over the course of the 17th century, New Mexico cost the crown 2,390,000 pesos, while Florida cost three times as much. The colonies were worth the expense as it was thought missionaries could pacify lands at less cost, and with more lasting impact, than soldiers. Franciscans in New Mexico were given an annual lump-sum payment from the viceroy for purchases needed in the missions, and caravans of 32 oxcarts loaded with good arrived traveled the Camino Real up to New Mexico every three years. Florida was resupplied more often, but both colonies were poor frontier outposts on the northern extreme of an expansive empire.
In New Mexico along the Rio Grande and in Florida, the sedentary agricultural populations provided missions with an existing population base to proselytize and draw labor from to run the mission. Franciscans would integrate themselves into communities and engage Amerindian help in building conventos or churches. Between 1610 and 1630 Franciscans built 25 missions in Pueblo communities and established 4 missions among the Zuni by 1629. The Franciscans arrived in Florida in 1573, and one hundred years later a string of missions, organized into four mission provinces, stretched up the Georgia coast and along the northern frontier of Florida, into the panhandle. Though covering a wide geographic range, the pure number of missionaries operating in New Mexico and Florida was small and usually hovered at slightly less than 50 friars. In 1655 70 Franciscans ministered to 26,000 converts in Florida, and in the early 1600s Fray Alonso de Benavides claimed Franciscans baptized 86,000 Apache, Navajo, and Pueblo converts (likely an inflated figure).
At the initial entrada, Franciscans attempted to dazzle Native Americans with vestments, music, paintings, ceremonies, and win over the population with gifts of bells, beads, metal tools, food and clothing. Accepting gifts created a sense of obligation and reciprocity, creating at the very least a trading relationship and building confidence for future interactions. Once friaries or conventos were established in key populations centers (cabecera or head of the mission system), the friars extended their influence by touring through visitas in smaller, nearby communities. In 18th century Texas and California the absence of large permanent Native American population centers required a policy of reducciones, or “reducing” dispersed natives into central towns before initiating spiritual instruction. Regardless of location, the friars attempted to build relationships with the Native American leadership as a way of winning over the lower class once their “natural lord” converted, as well as directing teaching at children who were viewed to be more malleable than adults.
At the missions adults and children learned the basics of Catholic doctrine, prayers, hymns and answers to catechism. Franciscans attempted to learn Amerindian languages, and provide bilingual catechisms and confesionarios when possible. One example, a Castilian-Timucuan catechism and confessional from Florida, was published in Mexico City. These types of texts, which start to appear in 1612, constitute the earliest surviving texts of North American Indian language. Because Christian doctrine and Christian lifeways were linked in the Spanish mindset, new converts needed to “live in a civilized manner, clothed and wearing shoes…” and were taught how to integrate European crops and animals into existing agricultural practices. Friars banned the corruptive influences of previous Native American religious traditions, including the ball game in Florida and kachinas in the pueblos. The friars also sought to protect mission populations from the corrupting influence of Spanish colonists and soldiers. By law, mission Indians could not travel to Hispanic towns, like St. Augustine or Santa Fe, without a pass and non-Indians could not live in, or spend more than 3 days, in Indian villages. Escaped neophytes would be hunted down, and corporal punishment was a common means of dealing with transgressions throughout the northern frontier for both Spanish and Native Americans.
The degree to which Native Americans adapted, re-interpreted, and modified the Spanish Catholic influence for their own purposes varied across the continent. In New Mexico, religious oppression forced the construction of kivas hidden under residential areas. There, outside the friar’s influence, traditional religious practices could continue in place of, and often alongside, participation in Catholic rituals. The missions provided a small measure of support and protection from illegal Spanish slaving raids, or from other hostile Amerindian nations. The Pueblos hoped for Spanish help and protection against Apache and Ute raids, as the Apaches would later flock to missions for protection against Comanche raids, and throughout the greater Southwest all sought protection from slave raiders attempting to find new souls for the mines in Northern Mexico. Slaving raids out of English Carolina would eventually bring about the downfall of the Florida missions as the sedentary populations made easy prey for slavers who nearly depopulated the peninsula by the early 18th century.
The constant pressure between Native Americans and the Spanish friars, soldiers, and colonists permeated all relationships in the colonies. Native American populations always greatly outnumbered the Spanish presence, and could respond violently to oppression. Perpetual tensions surrounding the control of Indian labor in 17th century New Mexico eventually created a violent response. Franciscans used Pueblo laborers to till fields, build churches, tend livestock, and perform domestic chores around the mission. Spanish colonists, entrenched in the encomienda system, tried to encroach on Pueblo land, and force the inhabitants to work their estates, pay tribute, and gather products for export south to Mexico. The tensions between the church and state over control of labor, as well as executing policy in the colony, exacerbated the smoldering hostilities between the Franciscans and Pueblo surrounding the continuation of native religious traditions. These tensions, combined with drought and the failure of the Spanish to protect the Pueblos from Apache raids, would eventually boil over to outright rebellion during the 1680 Pueblo Revolt.
For more info check out
The Spanish Frontier in North America by David Weber
The Early History of Greater Mexico by Altman, Cline, and Pescador
Spain in the Southwest: A Narrative History of Colonial New Mexico, Arizona, Texas and California by John Kessell
Pueblos, Spaniards, and the Kingdom of New Mexico by John Kessel
Children of Coyote, Missionaries of Saint Francis: Indian-Spanish Relations in Colonial California, 1769-1850 by Steven Hackel