Switching sides in sheep showmanship is a show process that is only done in some areas. Before you go to deep into switching sides or manuevering around your animal, know if that is a show process done in your area. In California switching sides and manuevering around your animal is a big part of showmanship and demonstrates your ability to move around and handle your animal perfectly.
To switch sides well and smooth, you need to begin your move once the judge switches sides on your animal and move smooth, calm, and swiftly to the other side. In order to get to the other side quickly you want to take your very first step with your inside leg, which will be the leg closest to your animal. By taking your first step with this leg, you reduce the number of steps you need to take to get over to the other side of your animal and in that way, you move more quickly to where you need to be. This allows you to get there sooner, get there smoother, and not be rushed.
You do not need to move fast, just smooth and steady. By taking your first step in this way, you appear to get to your new position faster, but you can move slower and smoother without looking rushed. The key is steady and smooth, so it is a win-win.