We've done it both ways. The one time that we did not take our own car seat and rented one from the car company, I was extremely disappointed. It was basically a hard plastic seat and much too small for our 3 year old. If your child is small, I would highly recommend you take your own seat and use it on the plane. If your child is not small, I would recommend that you check it as checked luggage. I'm cheap, so I don't have a special carrier. We usually fly Delta or Southwest. Both airlines pack the seat in plastic and we've never had a problem.
If you take your own carseat, you'll save money, your child will be more comfortable, and you will feel better knowing your child is in a safe, clean seat.
If you're worried about carrying the car seat, many times I've used our umbrella stroller to lug the car seat and let our child walk in the airport. He was ready to move around when he got off the plane.