I'm sure it varied from tribe to tribe and from region to region. But firsthand account I have read went along these lines:
Catch a horse by whatever means possible. If it's a leg and the leg breaks, there's dinner. Whatever. Tie the other end of the rope to a tree, and wait. Let the horse fight and wear itself out. After a few days tied up with no food and water, it should be suitably ready for breaking. One guy jumps up on the horse, a club in each hand, beats it to make it move off, and steers by means of clobbering either side of the head with said sticks. Sometimes they'd take a mare and foal for long rides. When they got hungry, they'd butcher the foal and carry its carcass along for substenance.
The cowboys would just tie it down, saddle it up, and buck it out. You can still see this with the Argentinian gauchos:
Video:
Gauchos in Argentina - Video