To upgrade pets you need pet gems which can be found in treasure chests or you can get them from guild auctions.
The reinforcement is for the pet basic stats. Each reinforcement adds 10% to the selected stat. E.g. for the pet above, the str will become 80% after a str inforcement.
The sum of a pet basic stats is an important value that reflect the quality of the pet. We call it the pet quality here.
Each pet get a fixed initial stats. The better the pet, the higher initial quality of the pet.
All pets get a same quality upper limit, that is 800%.
The quality for each pet is divided into 10 grades. Grade0 for initial quality, Grade10 for the full quality. Each grade represent half star.
For the pet above, the initial quality is 340%, the full quality is 800%. It need (800% – 340%)/10% = 46 times successful reinforcement to become a full quality pet.
So not every successful reinforcement will increase the number of stars.