IF YOUR PET IS LOW ON HITPOINTS (HP):

Visit the Faerieland Healing Springs and the Water Faerie there will try to heal your pets, or she may even give you a healing potion you can use. You can also feed your pet Healthshrooms to restore a few hitpoints.

IF YOUR PET HAS AN ILLNESS:

First visit the Neopian Hospital to find out what medicine your pet needs. Then, if you have a fast computer, you can try to buy the medicine out of the Neopian Pharmacy. If not, search the medicine on the shop wizard and buy it from someone's shop. Also, keep in mind that a Yellow Healthshroom will cure your pet's disease, so if you can get one for less than the price of the medicine you need, buy that instead and save neopoints.

If you can't afford to buy the medicine you need, try visiting the Faerieland Healing Springs. Eventually the faerie will heal your pet of it's illness. You can visit every hour or so, so keep trying.