The answer is yes! Indeed, you can bring your dog or cat along with you when you move to the Philippines.
The procedure is actually quite simple. First, you must take your pet to a USDA certified Vet in the States. Your vet must certify that the pet has been disease free for the past 30 days. There is paperwork that must be filled out, which you can obtain from the nearest Philippine Embassy or Consulate.
After taking care of these simple requirements, your pet is ready for the journey! As a matter of fact, there is not even any quarantine requirements on the Philippine end, so upon arrival, you and Sparky can be together right away!