M.F.
Any airline will let you gate-check your stroller and carrier-style carseat for free. That means you can take the stroller and carrier with you through security and leave it on the gangway right before you walk onto the airplane. I would recommend doing this as you won't have to pay to check it if your airline charges for bags.
You will not be able to take any kind of stoller on board as a carry-on. You would have to gate-check your stroller no matter what.
The sell bags for your stoller, but I've never used one. Never thought it was necessary and I didn't want to have to take the time to bag the stoller in the middle of the rush to get onto the plane.
Check with the airline you're flying on-Southwest lets everyone, including the child, check 2 items for free. If you're flying with Southwest, you might want to check your carseat/base together. In that case, I would buy a bag for the carseat, as it will have to ride the conveyor belt and all straps will need to be secured. I also know that on Southwest, if the flight isn't full, they will let you take a seat for free for your child to remain in the carrier. It only happened to work out for us once and the gate attendant offered it to us or we'd never have known about that.
I would definitely bring your own carseat/base and not rely on borrowing one from the rental car agency. It's always better to use what you know and what you know is safe, in my opinion.
If you're flying with your child on your lap, you will need to bring the baby's immunization record as proof of age and provide it to a representative at the ticket counter (NOT AT THE GATE) so your child can be added to your boarding pass. Those lines are usually pretty long, so leave LOTS of time in advance to get this done.
Other than the logistics, I can only recommend dressing the baby warmly, in layers, as planes tend to be cold. I'd bring a blanket, some toys, some books and a comfort object for your baby.
Good luck!