There are three simple steps on your way to get a MyPet.
Find a mob that is leadable is the easiest step. All leadable mobtypes can be found here.
This can be the hardest part but how hard it really will depends on the configuration of the Server.
Every MyPet-Type has a setting called LeashFlags that sets the requirements needed for the MyPet-Type to become a pet.
When Step 1 & 2 are done you just have to hit you target with the LeashItem (String by default). After hiting your target it will convert from a normal mob into a MyPet and you will see a message like this: You took a pet on a lead, it will be a good MyPet.
Congratulation you now have a MyPet.