I have been asked this question a lot of times, thus the reason I decided to share my experience with you. It can be a big hassle I must say.
We need to approach this question in two different ways: to include it in your check-in luggage or to bring it onboard.
If you have a small child and you need to bring your car seat with you in your travels, you can include it as check-in baggage. This is the simplest, most hassle-free way of flying with a car seat. The good news is that all airlines allow car seats as checked baggage with no additional cost. However, you should not forget about the dimension limitations of bags checked in for free. To be on the safe side, I would recommend buying a small child seat that you can bring with you during the flight.
The other way to fly with a car seat is to bring it onboard. This, however, may not be as easy as it sounds. Firstly, airline companies only let you bring FAA approved child safety seats on board. Secondly, there are many reports of flight attendants not letting parents bring their car seat onboard even though it was FAA approved.
My recommendation would then be to include the car seat in your checked baggage. This is the best way to avoid any headache.