First of all, we highly recommend to purchase a soft carrier bag that is explicity designed to transport pets. Have your pet sniff out the carrier bag and jump in and out of it. Implanting their smell in the bag means it isn't a new and unfamiliar item to them come flight day. If you don't own one already, buy a secure harness to put on your pet when you have to take them outside the bag for inspection or any other reason during the day's proceedings. 24-48 hours before your flight departure, feed your pet only half of their regular dietary intake. Try and not to show any feelings of stress or anxiety as this will likely be noticed by your faithful pet and make them tense as well.
On the day, place your pet in the soft carrier bag designed for animals. Place inside the bag a couple of toys that they love and even a t-shirt of yours to calm your pet. If your pet is not a fan of car travel (if you are taking this option to get to the airport), keep telling them soothing words and pat them through the bag. Commenting 'Good Boy' or 'Good Girl' is very reassuring for pets in an unknown environment.
Upon arrival at the airport, if you own a dog, take them out of the bag and allow them to stretch their legs a bit. Please double check your pets harness is stead fast and secure in case your pet becomes spooked and bolts. Have them sniff around a bit and even relieve themselves as this is an optimal way of reducing tension and nerves.
When you are settling into the plane, place your pets carrier on your lap and hug it tightly. If possible, try and allocate yourself a window seat so that your pet can look out the window to distract them. The hustle and bustle of people storing their luggage in overhead compartments and moving down the aisles can be frightening for your pet. Again continue to reassure your pet by patting them through the bag. Make them aware you are there and that they have nothing to worry about.
You will be instructed by airline staff to place your carrier bag underneath your seat or the one in front of you during take off and landing. Your pet may become distressed but this is regulation and cannot be altered.