Occasionally, creatures may drop eggs, which can be crafted into incubator. Once placed, a timer will appear over the incubator, showing how many more minutes it will take for the egg to hatch. Once the timer is up, the egg will vibrate rapidly. Slapping it in this state will cause a baby creature to appear in the incubator, which can then be named. To rename your pet you need a Memory Towelette which is made in the Hearth.
When a creature is tamed it will rest in its incubator, waiting for a command. You can then do 3 actions to your pet.
"Embiggen" your pet to make it, you guessed it, bigger and stronger. Each creature has its own Creature Catalyst to cause the growth. Essence must be collected from the bigger breed. Such as to make a Wompit grow, you will need Sterling Wompit Essence. Along with the Essence you will need various items to craft such as flatstone.
"Walk" your pet to have it fight by your side.