With most stuff, I just say whatever works for you, but in this case, you really should bring the car seat. Turbulence can happen at anytime and an unsecured child can be seriously hurt.
Here's what we do:
1- We "rent" one of the metal luggage carts to use. This should hold most everything, and get you to the baggage check in good shape. You can then either ditch it, or use it for your carry-ons and car seat, all the way to the gate. Some airports don't even charge for the carts.
2- "Gate check" the stroller.
3- Make sure you pre-board. Its much easier carrying the carry ons and car seat if the aisles are empty, AND there is no danger of tagging somebody in the head with it as you go by.
4- If you are by a window, the child gets the window seat. If you are in the middle section, the child cannot go in the aisle seat. The flight attendants should tell you this, but if you know it ahead of time it might save you some aggrevation. The reason for this is to keep the car seat from hindering an evacuation.
5- After landing, wait for everyone behind you to get off before trying to exit the plane. Again, its so much easier to carry everything off with the aisles empty.
6- Before getting your luggage from baggage claim, get/rent a metal cart.
Hope this helps!
(all of this is assuming you purcahased a seat for your child. If not, you will only be able to bring the car seat on if the flight has empty seats.)