Find a mob that is tameable is the easiest step. All mobtypes are tamable except Shulkers because they can't move.
This can be the hardest part but how hard it really will depends on the configuration of the Server. Every pet-Type has a setting called LeashFlags that sets the requirements needed for the pet-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 pet and you will see a message like this:
You took a pet on a lead, it will be a good pet.