Express Mail in USPS approved shipping boxes. Peafowl would need the filtered boxes (chickens don't need that) which are a bit more expensive. The Horizon box website
http://www.hm-e.net/ gives you a lot of information.
You need to check with the post office several days before you want to ship as your birds have to have a reserved "seat" on their flight. You can't just show up at the post office with the birds and expect them to take them.
There are temperature regulations as well, they can't go if any point of their flight is over 85ºF or under, I think, 20ºF. Not sure on the lower temperature as it's been awhile since I shipped any birds. If you are shipping birds from NYS to WA state, for example, and the temps look good across the northern tier of the country, be aware that they might be making a connecting flight in Atlanta where it's 100ºF on the tarmac. The pilots have the ultimate say on whether or not they'll take birds because of temperature.
As far as prepping the birds, I've never worried about that too much. You don't want them stressed the few days before shipping, though, and be sure to include lots of quartered apples or the like in the box with the birds. This gives them food and moisture both. Some people will give them their last water right in the post office parking lot. But generally speaking they are only going to be in the mail for 48 hours, maybe another 8 beyond that, so a few apples per bird should be fine.
Jennifer