We have had animals flown through Continental's Pet Safe program. United has recently adopted the same program (I think United and Continental are the same company now?), though they have certain breed restrictions. :( The drop-off/pick-ups are really easy and the animals (rats in our cases) never seemed worse for wear on the other side. There is a 24-hour Live Animal Desk that will help you out and every leg of transport is climate-controlled (though I think there are still sometimes restrictions on what time of year you can fly animals in places that experience extreme heat). Plus a few major airports have pet hotels for longer trips, though I don't have feedback on these. I believe you do need a special FAA-approved kennel.
United Pet Safe InfoVideo about Pet Safe