If you don’t currently own the Magic Wand Effect. You can buy one in the Catalogue if you navigate to Clothing → Temporary Effects → Magic Want Effect. This will cost you 150 Duckets. Once you’ve bought it, open your wardrobe and select the 6th tab. It should look like a green man with a golden plus sign next to it. Your wand should be in this section. Once you’ve found it, click on it and click “Save changes”. Your avatar should now have the wand equipped.
Place your Golden Egg in a room and press the “use” button while having your wand equipped. Click it 12 times to hatch an animal!
Now that you have a creature, to you need to click on it until it evolves. The different creatures and their evolutions can be seen below.