This might not be relevant for everyone. I flew in July 2012 with a cat from Vancouver, Canada to Jakarta. From what I've read here, the process seems pretty similar for cats and dogs. It might be different for other, less common animals, but I wouldn't know much about that. Also, there are some Canada-specific steps.
A little about our situation... At first we only wanted to fly back to Jakarta to attend a wedding, but one depressingly rainy and cold Vancouver afternoon we decided to move on a whim. With a flight booked in two months, we had to rush through all the moving preparations. When we took the cat to the vet for the first check-up, we only had 1 1/2 months to go. It was enough time, but only barely.
Here's how I would do it differently, step by step, if I could go back in time:
1. As soon as you decide you're moving to Indonesia:
-Contact JakPetz or Groovy Pets (agents). Get a quote on how much it would cost for them to take care of the paperwork in Indonesia. I asked both and JakPetz gave me a much lower quote than Groovy, so I went with them. Expect to pay 7.5 juta to 10 juta for this service. I know this seems like a lot, but it's a necessary expense. There's no way I could maneuver myself through the confusing Indonesian pet import system, and I'm Indonesian. For the price, they prepare an import permit for you, take care of the paperwork once you arrive in Indonesia and drive your pet to your home at the end of the quarantine period (there's an option to forgo quarantine, too). When my cat got home, she was clean and freshly bathed, although there was a lot of matted fur, but that's to be expected with a longhair cat.
-Get a copy of the Health Certificate form. In Canada, you can download this from the Canadian Food Inspection Agency website. If you're not in Canada, ask JakPetz/Groovy Pets which government body issues the Health Certificate in your country.
-Do your research before booking if you want your pet to fly as comfortably as possible. I didn't have the chance to find out a whole lot about this because we booked the tickets before we decided to move, so we were stuck with China Airlines. Contact various airlines to find out how they handle animal travel. For small animals, with certain airlines, you may be able to have your pet travel in the cabin with you. A forum member brought his cat from the US to Indonesia this way on Delta and it cost him $200. I flew on China airlines and paid $160 for the cat to fly as excess luggage. I was told some airlines like KLM allow you to see your pet at the transiting airport.
Make sure your pet carrier complies with airline regulations.
Also check with the airlines whether the country where you will transit requires an animal transit permit. We transited at Taiwan and had to get a permit. It was free and fairly easy to get over the Internet.
2. At least two to three months before departure (we did this 1 1/2 months before departure and just only made it):
-Bring the pet to your vet. Your pet will have to be vaccinated against rabies and some other diseases as required by the Indonesian government. Again, JakPetz or Groovy Pets should be able to explain which vaccinations are required.
3. One month after the first vet visit:
-Bring the pet to the vet again to get a blood sample for the rabies titre test (this cost me roughly $300). If you're in North America, the lab that will conduct the bloodwork is likely to be K-State University. The blood sample will probably be sent to the Kansas and sent back to your vet once the result is in. This will take roughly 3-4 weeks. If you're in a hurry like I was, you can contact the lab and order an expedited test for an extra $150 — they will try (but not guarantee) completion within 2 weeks. The test result will be sent to the vet's office.
4. At least two weeks before departure:
-Make a payment to the pet import agent. This is our deadline with Jakpetz. Other agents might have different deadlines.
-Request an Indonesian import permit from your agent.
4. Within five days of departure:
-Visit the vet one last time for the final check-up. Bring the Health Certificate form with you, as the vet has to fill the first few sections of it in. Collect the rabies test result now if you haven't done so.
-Visit an official government vet and get his/her signature on the Health Certificate form that your vet partially filled. Also bring any other documents required to export the animal (ie. rabies titre test result, import permit from Indonesia, vaccination certificate, etc. Check with your government agency which documents they want.) In Canada, you need to go to your nearest Canadian Food Inspection Agency office. You don't have to bring the animal with you.
5. Arriving in Jakarta:
-Make your way through immigration and baggage claim. Meet the pet import agent and pass all your pet's documents to him. He will go to the quarantine facilities and take care of the paperwork there. You can go visit the quarantine facilities with him; the process will take about 1 hour.
-After two weeks, your agent will contact you to notify you that the quarantine period is over and that they're delivering your pet to your address.
Hope this helps someone! Suggestions, additional information welcome — I'll gladly edit this post to make this guide clearer.