The ideal way is to keep your cat with you, but that does add significantly to the cost on most airlines, while some airlines do not allow pets in the cabin
Travel by car
The first thing is to make sure that your cat is comfortable and safe, so put your cat in a large sturdy carrier in which it can stand up and stretch and turn around easily. Cover the bottom of the carrier with some type of padding and secure the carrier with a seatbelt. If you are in a car accident, you want your cat to be as safe as possible, too.
If it is a short trip, under 6 hours, then your cat will be just fine but if it is a longer trip, you may want to let your cat out of the carrier periodically to drink water or use the litter box. Once you have arrived at
Travel by plane
There are two ways for your cat to travel by plane - in the cargo hold or in the cabin with you. The ideal way is to keep your cat with you, but that does add significantly to the cost on most airlines, while some airlines do not allow pets in the cabin.
Show up early at the airport to complete all formalities at your earliest. If you are, say, number three in queue among those with pets, and the airline allows only two pets in the cabin, then you will be forced to send your cat to cargo.
Be also sure that you have an airline approved carrier, and if in doubt, take it to the airport well before