Pet pieces for advancing combat pets?

I'm really confused how combat pet advancement works. Once I get a pet exploration level to 100, how do I change it to a combat pet? Do I need to pet pieces? And if I do, how many? Or is this referring to advancing combat pet levels once the pets fight alongside you? Also should I be buying my orange pet (Mistress) shards from the market even when they aren't discounted? Any clarification would be appreciated.