I've never taken my daughter's carseat on the plane (always checked it) and we have traveled by plane since she was 1. However, my daughter is very good about sitting in her seat. I've traveled a lot just myself and daughter and carrying the carseat probably would have put me over the edge. LOL!
I always take a portable dvd players (even for short trips) and something else for her to do when you can't use electronic devices and while you are waiting to take off. This can be books to look at, small dolls, coloring book and crayons. Look for small things that are easily stowed in the carry on and easily accessible.
One thing that helped me when traveling alone was a stroller. I used it in the airport mainly. If my daughter didn't want to ride in it, I could still load it down and it was always allowed through security. You do have to unload it, fold it down and put it through the xray, but it is worth it! Carts cannot go through security - only wheelchairs (with the person in it) and strollers. I used my stroller until my daughter was 4. If she was riding in it, I could hang stuff on the back (I took my big stroller when I was planning to actually use it during our vacation and the umbrella stroller when I was only planning to use it at the airport and when she was older).
Bring a snack or two for each and sippy cup for the little one. I have found that some sippy cups leak during take off because of the change in pressure. You may wish to bring an empty cup and buy a bottle of water before boarding the plane. Sometimes you have to sit on the plane for a while before take off and more wait time for the attendants to bring drinks around and a cup without a lid on the plane is just asking for disaster with little ones (and even big ones in my case!). A change of clothes for your little ones in your carry on is sometimes helpful.
Another thing - arrive at the airport / gate early. Give them time to see the area and burn off steam. Not to mention, if you are early you will be less stressed and that makes a huge difference. I always tried to look at it as an adventure and the times I haven't, well, they were not happy experiences.
My daughter likes to eat at the airport and that helps a lot. We find the gate, then get food from a nearby place and return to the gate to eat it. It is like a picnic. I try to make it fun for her and she is entertained while we are waiting. Don't get me wrong - she isn't starving, we either get a snack or meal depending on the time or if we are at a layover.
Hope this helps and enjoy your trip!