For horseback riding, you'll want riding boots if at all possible. Riding boots have a slightly raised heel, which helps prevent your foot from slipping in the stirrup and possibly getting caught.
If you invest in a good pair of riding boots, they can actually double as hiking books. An excellent brand is Ariat. They are not cheap, but they are well worth the price. I bought a pair of Ariats for a dude ranch vacation several years ago. They were so comfortable that I wore them for everything--hiking, BBQs, watching the rodeo, and even our day trip to Yellowstone.