Horses or yourself? Everyones levels are capped on 5 right now until more quests are added. For horses, training, showing, careers etc will level them up. 

I notice you have a lot of questions, which is of course perfectly okay but you can get some quicker answers by looking over the wiki which tends to have a lot of answers to your questions or you can message WW members like myself or mods, or for the quickest answers, ask on discord. You will likely get more help that way and a lot easier over multiple forum posts. 

Horses, and ty!