Horses are not native to the Americas, although there was an ancient species of horse that died out 11-13,000 years ago. The Spanish reintroduced horses to the Americas in the 1500s, but did not widely trade them. Horses have tremendous value as labor, transportation, and for war, and the Spanish were not about to aid possible enemies (or rebels) by making horses readily available to them. The English and French in Canada and on the east coast of the United States in the early 17th century had similar experiences.

Horses became more widely available after the 1680 Pueblo Revolt in New Mexico. The Revolt saw all the Spanish in New Mexico and Arizona pushed out of the territory (survivors from Santa Fe and the Rio Grade valley regrouped in El Paso, where they would remain for the next 12 years). They were forced to flee without most of their livestock, leaving behind perhaps 1,500 horses. The Pueblos are sedentary peoples, but are surrounded by nomads - primarily Utes, Navajos, Apaches, and Comanches (it's likely that Navajos and Apaches aided in the Revolt) - who quickly and effectively adopted horses into their cultures. Further trading and raiding quickly led to the diaspora of horses across the Great Plains, Rocky Mountains, and Western America, becoming so fused with some Native American cultures that it is difficult to conceive of (some of) them today without horse culture springing immediately to mind.

Sources: A Song for the Horse Nation, National Museum of the American Indian; The Pueblo Revolt: The Secret Rebellion That Drove the Spaniards Out of the Southwest, David Roberts

First Nations people started riding horses at different times, it largely depended on when the Europeans made contact with them. They were either traded with them, or given to the chief and his council in treaties.

They learned through the Europeans showing them, and in turn they taught the rest of the tribe.

This is a bit of a tangent, but is relevant to the topic

For the Shosone (show-so-knee) people, horses were very important. When the Spanish traded with them, it gave the Shosone an advantage over their enemies, the Blackfoot, allowing them to push into southern Canada. They were pushed back when the Blackfoot retaliated with guns recently traded to them by the English.