We flew (forget the airline) from Philadelphia to Atlanta last year and took Adalaid with us. They require a sturdy carrier and it has to be able to fit under a seat on the plane. Most of the smaller carriers with a metal frame, wheels and an extendable handle work fine (the sides can be soft).
Every airline is different - number of dogs allowed on a flight and cost. We paid $50 each way for Adalaid which I consider insane since our tickets were only $50 each for us. Some airlines charge over $100 each way for a dog in the cabin.
The best bet it to go to each airline's web site and read the details of what they allow and the cost.
BTW, Adalaid was a champ didn't bark once but she didn't like staying in the carrier the whole flight because she is used to sitting on our laps (she is spoiled
)