I don't know what can I do to cahnge my newly adopted pet. Can you help me, please?

Go to the pets page, click on the square dropdown thing for the pet you want to change, and there should be a "Change Name" option. It costs 500 CSC to do so.

Really?

Go to the main Pets page under the Personal tab. There's an option to change their name in the little drop-down menus next to each pet. It costs 500 CSC.

Edit: Sorry, when I started this post there were no replies yet. :P

Yep, there aren't any free name changes. Unless your account was inactive and someone claimed a pet name that you previously had, I believe you get one free name change for that.

Alright, thanks! :)