I live in Utah but ride a lot in South East Idaho. My wife grew up about 30 miles north of Idaho Falls. There are plenty of green mountains within a reasonible drive of Idaho Falls or Pocatello. Both cities have lots of farm land around them. This area of the state is much much colder in the winter than the other areas of the state.
This area is not far from Idaho Falls. Would be an easy day trip. The picture is in September and the green grass has already been frosted.
Down off the ridge tops, the grass hadn't been frost bit yet and was still green.
The trails are not as wide as those in the above post. But there are plenty of them to ride. But we often have to switch back and forth to climb over the passes.
We make several trips a year up and ride in South East Idaho. It is great riding.
I know a lot of the horse people in SE Idaho bring their horses down into Utah for horse sales and to compete in the World Championship Chariot racing etc. I don't know how much 4H and other kinds of stuff you find. But you will find lots of horse.
And from South eastern Idaho it's not too bad of a drive to come down for a weekend ride in the Canyon Country of Southern Utah during the winter months in Idaho, We have several friends who come down from Idaho Falls to join us each Jan-Feb to ride the desert areas.
Like this February Ride in the San Rafael of Central Utah
Or an April ride in Bryce Canyon