All the photos I have seen of Native Americans on horseback show the horse fitted with European or American style bridles and bits. When did Native Americans first start taming/training horses and what sort of tack did they use before contact with settlers and ranchers?
Horses were introduced to the Americas by settlers and ranchers, so there were no horses before contact with Europeans.
To make a long story short, they didn't.
Typically, the first time natives saw horses, it was in conjunction with Europeans. This is because horses are not native to North America. Instead of trying to design their own tack, they used the stuff that was readily available.