From one of our volunteers with experience being bitten by reptiles and birds:
1. Buy: large size bandaids, including knuckle and finger bandaids. Bottle of Bactine (tm) antiseptic. Tube of triple antibiotic ointment (like Neosporin (tm)).
2. When bitten:
- remove bird from the affected appendage 
- wash aching, bruised, and or bleeding hands with soap and water. If you do it quickly enough, the pain won't have set in as much.
- spray Bactine on the wounds. It has Lidocaine in it and will immediately reduce much of the pain.
- Bactine will slightly increase the bleeding for a minute, but unless you're a hemophiliac, don't worry. It will help clean out the wound.
- After bleeding has slowed, clean and dry wounds, apply triple antibiotic ointment, and apply bandaids over ointment.
- replace oitnment and bandaids until wounds have healed. I think the ointment reduces scaring.
3. consume a glass of wine or two. :)
Being sure you have an up-to-date Tetanus booster is also not a bad idea in general if you're working around animals.