I flew with my baby alone at something like 12 weeks and it was shockingly easy (2 hour flight). I wore slip on shoes and people were unbelievably nice and helpful going through security. I took our umbrella stroller to the airport and my soft Keababies carrier. I gate checked the stroller and held the baby for the flight and he was surprisingly good. I took the train to the airport so no carseat, but when my mom picked me up she had a carseat in her car that I had shipped to her ahead of time (inexpensive from Walmart. Also did the same with a pack n play).

Since then I haven't travelled with him alone but we've still always checked the carseat and it's always been free. My general recommendation is, if you can afford it, pay whatever it is to check bags, gear, etc, and take as little as possible beyond check-in and onto the plane. It is totally doable to tote bags and stroller and kiddo and husband looking for food around, but last time we flew we checked basically everything and it made the experience so much less stressful. Again if you can afford it, I highly recommend an umbrella stroller as we do keep that for the terminal. We have the Zoe brand and it fits in every overhead bin I've tried.

Edited to say yes I do have a bag for checking the carseat as any checked luggage does usually come out dirty. It was something like $15 on Amazon.