Contrary to popular opinion, airlines do not require tags hanging from your dog. Check with the airline you will be traveling on, for their specific requirements.
For travel in the United States with your service dog:
Notify the airline ahead of time that you will be traveling with a Service Dog. It is not required, but can help make your flight much more pleasant. Do not depend on your travel agent or anyone else to do this for you. The airline may be able to arrange your seating to allow for more leg room, or put you next to a vacant seat. They are generally extremely accommodating.
A Health Certificate is required for shipping in cargo, and for emotional support animals, but not for traveling with your service dog. Always carry proof of current vaccinations, etc.
A Prescription is not required for a trained service dog. A letter from your Doctor stating that you have a disability as listed under the ADA is sufficient. It never hurts to carry one just in case.
If you get separated, it is a good idea to have a personalized dog tag. The tags can say "Service Dog", list your cell phone, and home phone number. You can make these yourself, for just a few dollars at almost any place that carries pet supplies.
Micro-chip and register your dog! In case your dog slips their collar, or someone removes it, a micro-chip is forever. Animal Shelters and Veterinarians have the ability to check for micro-chips.
Vests & ID are not required, but make it easier for the staff to identify your dog as a service dog. We use a bandana with an ID tag, and when traveling may put on a stability harness. It depends on your individual needs.
Carry Identification, a picture of your dog, micro-chip and tag numbers. You can tuck similar information into your dogs pack. Your picture, cell phone number, emergency contact numbers, and your dogs Veterinarian's number.
Take a blanket to protect your dog from the temperature of the floor in the airport and on the airplane.
Prepare your dog:
Teach your dog to become "invisible". They should be able to settle down quietly beside you, or at your feet, and stay there for an extended period of time (hours). Take a road trip. If your dog will not fit or stay in the space at your feet, you may have to pay for an extra seat.
If your dog has been well socialized, the airport and flying should be no problem. If not, take them to the airport and introduce them to the sights, sounds, and smells. Take them on elevators, escalators, up and down stairs, busses, trains, over bumpy roads, anything that moves and makes noise.
You have to know that your dog will be inside the airport and on the plane for a very long time. Most layovers are not long enough for you to leave the security area, go outside and clear security again.
Teach your dog to "go potty" on command. This makes travel time much easier. Most dogs will get a little excited, with the activity of getting ready for a trip, make sure you see them go potty before boarding the airplane. Even if the airport says they have a place for the dogs to go potty, it is probably a little room. Most well trained dogs will refuse to go inside, no matter how many dogs have used it before. Some dogs will use a potty pad.
Pre-surgical rules work well. Withhold all food after 8:00 the night before if you will be flying early in the day. You do not want your dog to get dehydrated. Do not give excessive amounts of water, but small drinks on the trip. Whenever they bring you water on the plane, ask for 2. My dogs love ice water, or bring a small water bowl. You can't bring liquids into the airport, you can however bring a water bottle and fill it once through security. If you run out ask a flight attendant for water. Once you have landed give them all the water they want and of course a chance to go potty.
Plan ahead and use good sense, if your travel will take longer than is reasonable for your dog, make sure they get a chance to relieve themselves somehow. You may need to go outside, and back through security.
You should always carry supplies, poop bags, absorbant towels, and wipes so you can clean up after your dog, inside and outside.