Flying with an animal is two separate jobs: paperwork and the journey itself. Either can go wrong, and both are solved in advance.
A month before
Check the rules of the specific airline and the specific flight, not "airlines in general". Everything varies: maximum weight including the carrier, dimensions, whether cabin travel is allowed, how many animals are permitted per flight, whether prior approval is needed. The animal's place is booked separately and does run out.
Check the destination country's requirements. Often you need a microchip, a rabies vaccination given no earlier and no later than a set window, and sometimes an antibody titre test that takes months. Order matters: in many countries the vaccination only counts if given after microchipping, not before.
Two weeks before
Go to the clinic: examination, confirmation that vaccinations are current and recorded in the passport. Ask where and when to get the veterinary certificate — it has a short validity, usually a few days, and is issued just before departure.
Buy a carrier that meets the airline's rules. The animal must be able to stand at full height and turn around. Start acclimatising from that day, not at the airport.
The carrier
Leave it open at home with bedding and something that smells of home inside. Feed beside it, then inside it. Close the door for seconds, then minutes. Take a car trip with the door closed.
An animal for which the carrier is its own place handles a flight incomparably better than one shoved into an unfamiliar box on the day.
On the day
No food for four to six hours before departure; do not restrict water. Give a good walk before leaving. Arrive earlier than usual: check-in with an animal takes longer.
Write the pet's name, your name and your phone number on the carrier. Attach copies of the documents to the outside in a plastic wallet.
On sedatives
Do not give sedatives on your own initiative. At altitude they behave differently than at home: they suppress breathing more strongly and interfere with temperature regulation, and nobody is watching an animal in the hold. Many carriers explicitly refuse sedated animals. If you believe medication is unavoidable, that is a conversation with a vet, not a decision based on advice from a group chat.
Who is not accepted
Flat-faced dogs and cats — pugs, bulldogs, Persians, exotics — are refused outright by many airlines or accepted only in the cabin. The reason is serious: they struggle to breathe even on the ground, and pressure changes and heat in the hold are dangerous for them.