There are some things you should know in order to properly raise and care for a pet.
First of all, there are many kinds of pets. Some require a certain number of your acquired experience points, while others use only what is expended in battle. You won't have any problems raising pets if you can separate these two.
If you do not resurrect pets within 20 minutes of their death, they will lose their lives forever. If a pet dies, you should resurrect it immediately.
Pets consume pet food. If they become hungry, they will not show their normal level of activity.
If the hunger gauge falls below 50%, pets become very hungry and will suffer a decrease in movement and attack speed. Also, hungry pets cannot be returned to the condition in which they where in before they were summoned.
Pets whose hunger gauge falls to 0% are completely starved. They will lose faith in you and soon leave. The bottom line is to always ensure that your pets are sufficiently fed. Keep their hunger gauge at 10% or more to prevent them from leaving you.