The answer to this question depends in part on the time period. The word “established” suggests that you are referring mainly to the post-Conquest era of the 16th-17th centuries. French and Spanish colonizers created separate national spheres of influence in the Americas but also vied with each other to control some of the same people and territory. Their efforts and interactions were affected by the political structures of each European nation and of the Americas’ diverse indigenous polities, all of which changed over several centuries of colonization. France and Spain were both monarchies, whose rulers in the 16th century came from different dynastic houses. Both were Catholic countries that supported Indian conversion to Christianity. Religious beliefs included ideologies of European racial superiority, which they used to justify enslaving Indians and later Africans.
Geographically, the territories in and around the Caribbean Sea and Gulf of Mexico composed one large area in which French and Spanish forces competed. Farther north, the French expanded westward through what is now Canada and the Spanish through what became Central America, Mexico and the U.S. Southwest. The long Mississippi River basin was a contested are as the French pushed south and the Spanish pushed north.
In the Caribbean, many of the indigenous Arawak and Taíno people perished in the violent conquest or from disease. While the European powers claimed distinct islands for their respective nations, populating them initially with military forces and establishing bases from which to extend onto the continent. As the indigenous population declined drastically, indentured, poor whites arrived as colonists, while the remaining Indians were more often enslaved. The French and Spanish divided the island of Hispaniola between them, with the French half becoming contemporary Haiti and the Spanish half the Dominican Republic. As plantation agriculture intensified on the island, imported enslaved Africans soon became the primary labor force.
In the northeastern U.S. and southeastern Canada, primarily what is now New York State and Ontario province, the French encountered the Haudenosaunee or Iroquois Confederacy, an organization of five Indian nations. For decades they violently opposed the French, until a group of Mohawk broke off and sided with them. In this area, the French used Indian alliances in their conflicts with the British colonizers. In most of the French-controlled northern territory, however, the indigenous Algonquian-speaking peoples’ political organization was in tribes or chiefdoms so that French official policy was put in to practice on the ground in ways shaped by each group’s different features, as well as influenced by the priorities of different religious orders. Jesuits, for example, learned indigenous languages in order to catechize.
The French expanded, while engaging in constant skirmishes, primarily through a combination of negotiation, conversion, and trade; they continued to establish outposts, many of which combined forts, missions, and trading posts all-in-one. In north-central North America, in particular, the fur trade came to dominate Franco-Indian commerce. Through Michigan and then Illinois, the French continued west and then south, until LaSalle’s forces, moving down the Mississippi River to the Gulf Delta, claimed Louisiana in 1682. Important settlements in this vast territory included contemporary Chicago, New Orleans, and Mobile, its first capital.
Earlier, both French and Spanish explorers had encountered tribes and chiefdoms on Caribbean islands and the mainland around the Caribbean and Gulf of Mexico. The Spanish gained much greater experience with state- and imperial-level organization. They had landed in Mexico in the early 16th century and moved inward to contact and later conquer the Aztec Empire, then one of the largest polities not only in the New World but in the whole world. Their experiences with a more diverse array of political systems led them to develop a broader set of policies of interaction. The wars of conquest, together with the European-introduced diseases, had devastated the indigenous people. Spanish genocidal violence generated fear and mistrust among the Indians, including reluctant cooperation and violent resistance. Even as the practice of enslaving native peoples became standard, some Spanish clerics such as Las Casas called for more humane treatment.