How to get a pet

1. Find the mob you want

Find a mob that is tameable is the easiest step. All mobtypes are tamable except shulkers because they can't move.

2. Meeting the requirements

This can be the hardest part. Every pet-type has a setting called LeashFlags that sets the requirements needed for the pet-type to become a pet.

3. Taking mob on a lead

When steps 1 & 2 are done, you just have to hit (attack) your target with the leash item (lead by default). After hitting 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.

Using the lead as a weapon is required to perform the actual act of taming. Putting the mob on the lead will do nothing towards taming your new pet.

Alternatively, you can skip all these steps and purchase a pet directly from the pet shop with /petshop if you have enough money.