I've actually flown quite a bit with my son in his first year. Here are some things that made it easier for us.
--Whenever possible, fly Southwest. They do a number of things to make it easier for families including = not charging for bags under 50 lbs, allowing families with children to board no later than the middle of the flight, and encouraging their employees to be FRIENDLY to families. I was Premier or high on United for more than 10 years, and I've given them up to fly Southwest.
--We don't buy an extra seat for my son when we fly Southwest (they can always give you a seat if the flight isn't sold out because of their open seating) or when we fly shorter flights (1 hour or less). However, we just flew to CA on United and found it helpful to have the extra seat between us.
--If you can, choose a mid-day time to fly.
--Make sure you have a copy of your son's birth certificate with you at all times.
--We've never found having a car seat with us helpful.
--Check in everything and anything on line before you leave the house. Get your boarding pass, and if you are flying an airline that charges for baggage, pay those fees ahead of time. I don't know American's procedures, but if you do this with United, there's a special check in counter that takes seconds.
--Pack formula in pre-measured containers. Avent makes a bowl that holds three, and we get a lot of use out of Inno Baby Packin Smart containers. I pack empty bottles, take the formula powder, and buy bottles of water after I get through security. At that age, I would make sure you have two extra portions in case of delays.
--When I fly with breastmilk or food, I pack it in an insulated lunch bag and hand it to security before I fly through.
--If you need to carry on any ointments, medicines, etc. for your son, be sure to put them in a quart size ziploc bag and hand it to security as well.
--Be prepared to break down everything for security. Some agents even made me take off my son's hoodie and robeez on our last flight.
--Think about the flight time and your son's schedule. You may want to get to the airport early to make sure you aren't stuck in the security line during a nap or eating time.
--Pack as light as possible for your carry ons. You won't have time or hands to do anything anyway.
--When you get to the gate, ask for a gate check tag for your stroller or one each for your infant seat and snap and go. It's helpful to the check in staff if they don't have to stop the check in process to get these for you.
--When we go to the east coast, I do not change my son's schedule. It's actually kind of nice to have him sleep 8-8 instead of 7-7. However, when we just went to CA, we did adjust him to Pacific time. We aimed for a 30 minute change each day, and it worked out beautifully.
Most importantly, stay calm and take your time. It really isn't as hard as it seems as long as you accept that this won't be a speedy process.
Feel free to send me a private email if you have mroe specific questions.
Enjoy your trip!