Youtube has some good videos on it.
Pigeon wise I just take the breasts. Lop of the wings, get you fingers in under the breastbone and pull up and out, separate the breastbone from the bird, very little mess/effort (pros do it in like 10 seconds).
Rabbits, cut the skin near the anus/genitalia and cut off the heads and legs at the lower joints. Cut up the belly (just the skin, not the meat), and then peel the skin off the rabbit firmly (it comes off pretty easily). Be careful not to puncture or damage the digestive system. You can then remove the guts and fillet it, or brine it if you so desire.
ALWAYS check the health of your quarry before skinning, especially signs of mixie on the rabbits, look at the liver, fur and eyes in general, if it smells dodgy/looks weird or discoloured/has spots etc chuck it away.
Make sure you practice first (goes without saying) that you can hit the killzone (roughly and inch) with your rifle at the range you will be shooting, and you know where the killzone is.
Where gloves for skinning as well, have some water and a bin ready (learnt that the hard way...)
Lastly, stay safe. Know what's behind your target and the laws of what you're doing, and always observe safe firearm handling laws (even if it's only an airgun, they can fatally wound, even if it seems less likely).
