To change your home stable, simply go to the stable you want to set as your new home stable and click on the door. You will get a pop-up window asking if you want to set this new stable as your home stable in exchange for 25 Star Coins.
Please note that you need to be a Star Rider to change your home stable to certain locations and that your home stable will be automatically reset to Moorland if you take a break from being a Star Rider.
Click on the door of your new stable if you want to change it.