Travelling with a non-mobile baby is fairly easy, except for all the "stuff" you have to lug. The worst part is getting through security, after that it's pretty easy. Since the baby in non-mobile, I suggest cheking your stroller (if you think you need it) and carseat (which you will absolutely need) with your luggage. There are cases you can buy to protect both from damage (check out Amazon). If you are worried about checking those in with the luggage, you can haul it all through securtiy and gate check instead.
If you check-in your stroller and carseat before going thru security, make sure you have an Ergo or some other kind of baby carrier, so you can have your hands free.
For security - the stroller will have to be folded up and it and the carseat will have to go thru the scanner, so baby will have to come out and be carried thru the metal detector. I highly suggest wearing yoga pants and a bra without an underwire and as little metal as possible so you don't have an issue going thru the metal detector, some airport have them set very sensitively.
For baby's formula if you end up using it, I suggest getting the pre-made ready to eat (RTE) bottles specifically for the flights. No mixing, no worrying about buying or getting enough water while you are in the airport to mix, it's just done. Plus you won't have to carry around a bunch of dirty bottles. Just make sure you stick all those bottles in their own clear baggie (a gallon size is fine), and take them out of the diaper bag for the TSA people to look at. You can also have your typical 3-1-1 quart sized baggie with your other liquids and gels in it. They will wipe a bottle down and put the wiper in the chemical sniffer. They make also pull your DH aside for extra pat-downs, etc because of it, especially if the stroller doesn't fit thru the scanner. Basically, make you you get to the airport early enough to intend to have issues at security. You don't want to be rushed to get to your gate after you get out of security. Depending on your airport and previosu experiences with lines at security, plan at least an extra 30 minutes -1 hour than you would pre-baby days.
Your baby will be a lap baby (free) unless you intend to buy them a seat, in which case, you would bring your carseat on the airplane (just make sure it's FAA approved, most are). I have yet to run into an airline that will charge you extra for the carseat or stroller as part of your luggage. That is usually complimentary (amazing since they nickle and dime you for everything else).
We always take are regular carseat with us. I know some people just buy a different one for travelling, but we really haven't had any issues. Most we had was a tear in the stroller that didn't affect the function and some dirt on the carseat cover. Both of these were when we didn't yet have a protective cover for either.
For baby not crying the whole way, I truthfully have never had an issue (I am so lucky to have had the perfect, easy "starter" baby), the droning of the plane knocks my kiddo out cold. For take-off and landing, he either got a bottle or a pacifier depending on whether it was time to eat or not. It worked out well for us.
Must haves - make sure you have a change of clothes for everyone in your carry-on (in case of blow-outs), baggage check as much as possible so you don't have to drag everything thru the airport. Have enough formula for at least an extra 2-3 feedings to cover any potential delays. Same for diapers, but I usually have 4-5 extra. I also took some powedered formula and one empty bottle for extreme emergencies. Pacifiers (at least 2-3 in case they get lost). I highly suggest a clip for those so you can just attach it to the baby. They don't really need toys or books or anything at that point, maybe a small toy they can grasp and look at if they are doing that then, but don't over load.
One tip for the heavy formula RTE bottles if you go that way, once you get thru security, divide the bottles up into half between your diaper bag that will go under your seat and your roller which will go over head. It'll just make things lighter to lug around and little less bulky once you are on the plane. Same with the diapers and wipes. You just want your essentials in your diaper bag so it'll be easier to find stuff while you are flying.
Sorry for the novel, but this is what I found helpful.