United and Southwest (and I'm sure pretty much all the others) will allow you to check one stroller and one car seat for free per child, no matter how many bags you have, so don't worry about having to pay for that. Just get a car seat backpack to make it easy to get from the car to the ticket counter and then to rental cars. We just got
this one from Amazon and it was great this morning dropping my wife off at the airport. Super easy to carry it and push a stroller.
Never taken the car seat on the plane, but if you do it needs to say "This restraint is certified for use in motor vehicles and aircraft" or they won't let you take it on. For a stroller, I recommend just getting an umbrella style one and gate check it. Much easier to break them down and put them through the x-ray.