When treating a pet that has swallowed something, you can extract the item from the belly to get a free item. The next day when the owner comes to pick up the pet you get will get money and an item.
In the beginning you can only cook eggs. Just stick with it -- making harder food costs more items and is just as effective.
Open the cage that your pet is in, tap him, then select "Put in inventory".
If you treat Hilary Duff's pet well you will get more people to come over to you to treat their pets.