Native American tribes traveled by way of walking, dugout canoes and horseback. Horses are not native to the Americas, and many tribes did not have them until the 1700s.
Native Americans walked to get wherever they needed to go on land. When needing to travel by water they used dugout canoes. Dugouts were made from the trunks of large trees, usually cottonwood trees, that were "dugout" or hollowed out using axes made of stone and carefully constructed fires. These canoes were heavy and were propelled through the water using long poles.
Horses became popular when the Apache Tribe began to raid Spanish settlements for the beasts in the 1650s. The Apache traded the horses with other tribes resulting in the spread of the animal throughout the various Native American Tribes.