You have to wear all the clothing from a set at the same time. By far the easiest to get is the "pet" set of clothing. Choose the Pomeranian in survival mode. You get the Pet Collar, Pet Shoes, and Pet Clothes from completing the D, C, and B challenges that unlock once you've completed all the challenges from the respective tier.
I don't remember but I'm pretty sure you also get the Pet Hat through the first tier of Pomeranian challenges too, and if not, you can get it through a random item drop.