I've travelled a few times to breed and hunt tests in SC. You will need a travel health certificate and be up to date on shots. The cert you can get at your vet for up to ten days. Any longer and you will have to get one for the ride back. I highly recommend a good tick and heartworm med just prior to leaving.
The crate is inspected by TSA, and even though the airlines have their standards, TSA trumps them. The crate MUST be bolted together. Use stainless screws. Plastic is a no-no. Get a water/food dish that clamps to the inside of the crate door. Petsmart has some. You can't put food, water or toys in the cage, according to whatever mood TSA is in that hour. One time they required the food AND water dish. I asked what food was available since i couldnt supply it. . Crickets.
Give yourself an extra hour and be aware TSA at some airports are 9-5. So if you have an extra early or late flight, ask about their hours. I almost missed a flight in NC because they weren't on duty yet.
As for airlines.... ALASKA! They are the very best. They are cheapest as well and take far better care of the dog AND you. United is 2nd in my opine because they dont tell you when the dog is placed in cargo like Alaska, and they cost more. Delta has the most experience, cost the most, and seem to have more issues with complaints.
I personally witnessed my dog with Alaska a few times being loaded. The people handling her had no idea i was watching from the window. They used two people to move her in her crate, and were more than delicate. They tried petting her through the crate slats. She was more than happy. She (i had two different dogs for different times) did not suffer any anxiety, but my dogs and breed are an exception. Plan ahead, do homework and be prepared.