I don't know how to feed my pet

Click Petz at the top of the page then go to the market and buy food.

Go back to the Petz link and you will then be able to feed your pet using the links with each food item.

One of your pets has died, and another is starting.

After a prolonged time of not feeding them, their hunger increases while their health decreases...until death.
You may be able to Resurrect it.
If you cannot feed them as needed, you can put a fully fed pet into the kennel
They will stay at the same degree of hunger/fullness as when you put them in.
You can keep them there indefinitely, or pay 30 blessings per day to remove them.