I was wondering how you get your pet's proficiency to rank up. I know how it learns knew moves but I want to know how they can get their moves stronger. Thank you in advance!

It's a feature which has not yet been incorporated into the site. However, once it is implemented, it will work like this: After a pet has reached level sixty, it will begin to accumulate Expertise by battling enemies. As the Expertise Rating for a certain Ability increases, the Ability will eventually gain rank.