Hi Jana,
I have flown with Kookie before and after considering the options, its better if your grey flies with you in the plane, it means less time away from you. Basically most airlines will not allow you to take him along in the cabin, but they would allow you to take him as excess baggage. This means they weigh him in his travel cage at the checkin counter in the airport and you have to pay for it as excess baggage. This is mainly cheaper than flying him as cargo. He goes on the plane into a special room in the plane. It should be a pressurized compartment, where normally pets fly. You have to check with the airlines in advance to be sure the flight can accommodate pets. Also you have make sure he has enough food and water for the flight. The travel cage is better a plastic pet carrier, so that he cannot see through except from one side. This will make sure he is not scared if other animals are travelling along with him on the same flight. Check with the airlines about the carry box standards, they would give you the dimensions and specifications. Also make sure to have all the travel documents ready so that you will not face any delays at the airport upon arrival.
Normally for greys to travel, you need to get 2 cities permits, one is the export permit from your country of departure and the second is a cities import permit in the destination country.
they might also require bird flew test and a vet health certificate. You have to inquire about those.
Have a safe trip