I've been here for 15 years now and I still have yet to get a custom pet/profile/etc. I want to start with a custom pet, but where do I get one? How do I get one? What are the steps? D:
Having several custom pets of my own, I usually start with the basics of character design. What do these characters look like, what's their story, etc. It can be as detailed or as bare bones as you want, so long as you are happy with it. From there I move on to commissioning artwork and overlays from artists in the Art Market area of the forums. Once I have an overlay then I submit it for approval (select the "Custom Overlay" link at the top of your Pets page, then select the upload option. Overlays are 500CSC each to submit, though the credits will not be removed from your account until the overlay is approved).
You can find free pre-made pet profile codes in the Profiles & Graphics Market section of forums, or can commission more unique codes from other users if coding is not your thing. Likewise, there are some users who offer commissions for pet stories.
You can do these steps in whatever order you want, at whatever pace suits you. I've been here for 18 years and still don't have a 100% completed pet.

Thank you so much for your response! This is perfect and exactly what I was curious about, I appreciate it a lot! :)