If you want to catch a horse, you need to find a wild horse and use a "Capturing Rope", after using the Rope a minigame will start, you can give the horse "Lump of Raw Sugar" to have a higher chance of taming the horse.

You need ropes (buy them from a stable manager) and lump of raw sugar (craft them via heating 10 raw suger & 1 mineral water. Both can be bought from a food vendor). Search a horse, go near, throw your rope and press space bar. then dry to go closer - when it stirrs up press space bar until youre close enough. Press F5 and feed the horse 3-5 lump of raw sugar, then try to ride it via R.