I think the airline you use depends on where you are shipping to & from. I think Jill is correct in that Southwest does not allow pets on.. there is no room for the Sherpa under the seat and they do not ship. The main thing is that the plane has a pressurized climate controlled cargo area. Just call that airlines pet desk and let them make the reservations.
The last time I had a dog shipped, he was being shipped to me from Nashville to KCI. The only airline I could get him on was Delta. I don't remember the cost but he arrived completely unstressed and just perfect. This was a six month old puppy though.
My friend flew out to pick up her puppy and if memory serves, she flew either American or United. She flew him home in a carryon - which is the best way to do that.
On another note, if you do plan on shipping the puppy, I'd spend some time getting them used to being crated alone. This will go the farthest in reducing stress on the puppy.