You will be able to gain all types of pets from eggs that you can win through doing different in-game activities in the Golden Season Pass, and from Emerald Suitcases. When you win an egg you will be able to place it in the incubator. 

It will take 4 days for an egg to hatch, but if you’re impatient to see your brand new pet you’ll be able to speed up the process using diamonds.