Hey i just brough my 4 cats over from england with me. Its expensive.
They will need to be FULLY up to date on all regular vaccinations, wormed, fleaed. Then they'll need the rabies vaccination, and for this to be checked that its taken after a month with a blood test. And 7 days (no more) before flying they need a fitness to fly certificate from a vet (also known as certificate of health)
Then you'll need to look for an airline that allows pets, you can probably take a chihuahua mix on the actual flight with you as your hand luggage (just get a sturdy airline approved pet carrier) and they can go under the set in front of you. If not you will need to get a kennel (pref wooden) suited to your dogs dimensions so that they will be comfy, and then you pay the airline for them to have them in the hold.
At the airport when you land you state at customs that you had a pet on the flight and you will then need to pay immigration some money to check out your dog and make sure theyre healthy (about $150 for my 3 cats, $60 for one cat) Then you go down to the shipping bay where the animals are taken, give them your papers immigration gave you, then pay them about $60 for the pleasure of picking up your animal, then go back to immigration to give them more paperwork you will aquire! Then you're good to go.
If you're lucky immigration may bring your pet into the main airport when they check meaning the paperwork will be exchanged by them, but mostly its a lot of backwards and forwards! Have cash on you - they don't like accepting cards!
Have your proof of ownership with you, vaccination cards, microchip/tattoo number (if they dont have one, get one), and dont have a collar on them for the flight. Have the rabies (stamped) certificate and their fitness to fly certificate and any other vaccination things. You will also need to write a letter stating the name of your pet, the microchip/tattoo number and that they belong to you, the address they came from, vets address and address theyre going too.
Feel free to PM me if you have any other questions