To get a horse, it's two steps.
First you need the riding skill from the store, which you buy with your turbine points.
Then you go to the horse farm where they sell the actual horses. A basic mount is 200 silver, but the faster ones are 500 silver.
Once you buy one, you get something like a 60% discount on any future horses you happen to buy.
I like the Genkimono idea for my horse's name, especially since it's a red horse.
Too bad you can't whip it to make it go faster, haha.