I had to help with my parents' pets when they moved two states away. They moved, and then once the moving van arrived a their new home with their stuff, I shipped the pets on the airline to them. We used Delta, and it worked out great. I would recommend a non-stop flight, as transfers can be terrifying for pets (having observed some terrified dogs abandoned in luggage areas in my travels).
The way it worked was, you have to ship the pets as "cargo," and pay freight for them (this may work differently if you yourself are on the flight, but freight charges will apply either way, and it's not cheap). Anyway, you provide the crate. It needs to be a hard-sided crate, large enough for the animal to stand up in without touching the top, and big enough for them to turn around in. You need to put a water tray inside the crate (empty), and tape a bag of food (enough for 1 meal for your pet) to the crate. The pilot will be notified (and if you're on the flight, speak to him/her personally to remind) that there's a live animal in the cargo hold. This is critical, because if they don't turn on the cargo heaters and pressurize the hold, your pet will not live through the flight. Our vet recommended giving the pets Benadryl to help keep them calm and keep their ears clear (that was super fun with the cats, let me tell you).
I believe you can only bring your dog on the plane (in the cabin) with you on certain airlines, and only with a very small dog. I think a 50 pound dog would probably have to go in the cargo hold.
Check into which airlines have direct flights to wherever you're going, and then call them for further info on how they handle pets. Like I said, we used Delta and it worked out very well. They were very good with the pets and everyone got there safe and sound.
Oh - forgot to mention - you will need to provide a crapload of veterinary paperwork, as well as fill out a big stack of paperwork at the airport. Arrive at least an hour earlier than you otherwise would. It's quite a process.