Long trousers (jeans work, riding-specific pants aren't necessary) and closed-toe shoes with a small heel are the two non-negotiables — sandals or open shoes aren't safe in stirrups.
Beyond that: a warm layer even in summer, since mountain weather shifts fast once you're up in the foothills; sun protection (hat, sunscreen, sunglasses) since UV is stronger at altitude than it feels; a light rain layer if you're traveling June through August, when afternoon showers are common; and a small daypack for water and a camera, worn on your back rather than a shoulder bag that can swing around and spook a horse.
Overnight yurt tours need one more thing — a warm layer for the evening, since camps sit higher and colder than town. We'll flag any tour-specific extras when you book.