I brought cats to the US from Italy. They needed the rabies vaccine and it had to be within a certain time frame before the flight. I believe it was no more than three weeks and no less than on week, more or less but you should check. I found most of my information on line and then checked with the airline. You will probably need to go to a specific veterinarian for the final ok. They also needed to be microchipped. And make sure that your airline knows you will have a dog with you because some airlines do not allow animals in the cabin with you. (I hope you aren't putting your puppy in the cargo hold because animals have been known to die down there from extreme temperatures and fright.)
I brought cats to the US