Yeah, what AHappyChick said. You have to use a box with some sort of filter like a Horizon box (typically people charge anywhere from $10-$30 for the box depending on size and whether it is new). It is much better to ship them when it is cold and most people either put no food or water in the box or just fruit slices. Definitely don't do dry food. Do try to get them to the post office as late in the day as possible, so they will do the bulk of the transit over night. If you are able to find out, take them to one of the main post offices or an airport post office if you have that option in your city. The last time I tried to find out though which post offices the birds go to in transit before they get to our neighborhood office, the post office would not tell me because they said it is a security risk (understandable).
Generally though, the birds do fine if they are only in transit one or even two days. The cost varies a lot depending on what you are shipping. My last shipment was roughly $130 (plus the box charge), but that was for 8 bantam ducks. I have never had shipping be under about $55-$60 (in recent years). There is the option of shipping them by air as well, but few people do that with poultry/waterfowl because it is more work and more expense if you are only shipping a few birds. The buyer and seller have to go to the airport and generally deal with air cargo. Again though, I don't know anyone that is doing that with waterfowl anymore and you would have to be shipping several birds to make it cost effective.
HTH!