I've traveled alone with my son a lot, from infancy to toddlerhood. What I learned:
You probably have one of those bucket type carseats? Where the seat fastens into a base? Is it one that can strap into a car on its own without the base? (ours was a Chicco).
You will buy your tickets as an adult with a lap infant. Baby is free, up to age 2, if she is a lap child.
Different airlines cater to lap infants differently. Frontier automatically assigns the mom plus lap child to a seat where the seat beside you is vacant, if the flight is not full. Others will let you request a seat change at the gate, to a spot where the seat beside you is open.
Sometimes, then, they will let you put your carseat in that slot. Ask ahead and at every check, because sometimes it depends on the flight crew not the airline.
Because of the above, up until he outgrew his infant seat, I always brought him in it, to the gate. Then I'd bring it on the plane if they had a spot, or gate check it if they didn't.
Rental car seats SUCK. Expired, missing directions, mildewed and dirty... I'm appalled at what I've found. They also make you install it, and don't always have the instructions or specs on which is safe for your child.
A stroller is invaluable, in large part because there aren't luggage carts beyond security in most airports. I've tried to make it without a stroller and it's just an unnecessary nightmare.
Checking the stroller at the gate is free, too. As is checking the car seat at the gate or ticket counter.
Some airlines require ID for the baby, some don't. A vaccine record or birth certificate works.
And some make you go to the ticket counter even if you have no checked bags.
What else... Baby medicines don't have to be in your single 1 quart ziploc with fluids. You can have a second ziploc. If you bring milk for LO (I always bring a quart of soymilk for myself, lol, and say its for the baby), just allow a few extra minutes at security for them to check it (they just wave a thingie over it, they don't touch it).
It's good to say ahead of time that you may need assistance, if you have a connection with a layover under 2 hours.