The procedures are simple and cheap:
FIRST: Check with your destination the laws concerning flying in animals in winter.
1. Get a rabies shot at least one month but no more than six months before departure (put the microchip in first).
2. Get her a crate just big enough to sit up without their ears touching the top (about NT$5,000). Check with the airline that it's suitable.
3. Measure her weight and note the weight and dimensions of the crate, then book them on your flight with you; ask for info on how/where you collect them at destination.
4. Less than 10 days before departure, get a health certificate from a vet, then take her, the health certificate, your passport and flight details to the quarantine office near NTU Veterinary Hospital for an export permit (free).
5. Get her used to enjoying the crate.
6. Don't feed her the day of travel; arrive very early at the airport so you can take her for a long walk around before she has to go in the crate.
7. Check in with them; they will ask you to get the paperwork stamped at the airport quarantine office (free).
8. Return to check-in; the clerk will tell you how much her flight will cost (about NT$6,000 to NT$9,000 usually).
9. Pay the fee at the nearby fee counter.
10. Check her in; she will then be taken away to be put on the flight.
11. Tell all staff on your way onto the plane that you have dogs in the baggage hold (to make sure they turn on the heating); tell everyone, especially once seated so the captain can be informed, even though he will already have been told (best be safe).
12. Pick her up (ask where; sometimes it's near the baggage carousel if not actually on it); walk her through customs, and you're home and dry (well, check the crate [;-)]
Take her for a walk as soon as you arrive. Line the crate with pee pads and put in something like a rope toy that she likes. Have a water bottle on the crate for her, and teach her in advance how to use it by using meaty water.
Hope that helps, but feel free to write with more questions.