Definitely bring your carseats (for the 2 and 3 year old) on the plane with you! If you have your children on your lap and there is bad turbulance or a bumpy landing, then any kids on your lap become your air bag. Yikes! If they are sitting in the seats the 2 and 3 year old will be too small for the lap only belt and could slide out. Besides, a few years ago (before I knew better) I had our daughter on my lap for the flight and then rented a car seat with the car. Yuck, yuck. It was a really slimy, old, cheap seat that cost me about $65 to rent for the week. Our extra car seat for grandma was only $50 new. To transport both carseats you might want to try a luggage cart or a MicroMover. Usually with the correct positioning of the seats (one inverted on top of the other) and bungee cords, then both seats can be rolled to the plane together. With three kids, you'll have your hands full. For your oldest, they will not allow you to use a booster. No belt positioning booster seats can be used with a lap only belt and planes only have lap belts. For ease of transport, you might want to bring a backless booster. These are relatively small and easy to slip into your carryon luggage.
But please don't check your carseats with your luggage unless you have padded them very well in large boxes. If you are in a crash you need to replace your seats and many people compare the handling of the luggage to the forces that could occur in a moderate crash. Luggage is often tossed around. If you do not want to use the car seats on the plane, I would gate check them. At least this minimizes the handling.
Sorry this sounds sort of preachy, I just feel very passionate about child seats (ok, I'm a nerd!!) Have a wonderful trip!
Jennifer