I recently adopted the CAG from a bird rescue in Missouri, so I can only speak to my experience with them, as well as the 5-6 other rescues I have contacted. Here's how it seems to go:
1. Application, either online or a paper form emailed to the rescue. They typically ask what kind of birds you have or had, what happened to them, why you would want an exotic bird, if you are familiar with the care they require
2. Set an appointment to meet the bird.
3. Visit the bird, sometimes an in-person interview. The rescues want to screen out breeders and people who are looking for discounted birds to flip or scam folks with online. They want to see if you can handle the noise, the beak, etc. Don't expect to leave with a bird on the first visit.
4. Waiting period. Some of them have a requirement of going to a "bird class" to learn how to take care of birds. Some will require you to wait a few days to be entirely sure that you want that particular bird. I live 3+ hours away, so I emailed that I still wanted Smoky Blue.
5. Pick up bird. Some rescues have cages, some require you to bring one. I brought my own, with food, water, and fruit for the new guy.
6. Follow up. Some of them have "return policies"- if it's not working out, you are expected to return the bird to that shelter, rather than find a new home on your own.
Hope this helps