I have been doing this for the last few years (both international and domestic)

Followings are my priorities whenever i am with infant and i have never had any problem:

1. Carry an umbrella stroller (usually, its for 6month+, however, if you buy a good quality one, you can incline almost 170 degree)
2. You can carry stroller for free, no extra charge at all, but while checking-in the counter, you need to let the airlines staff about the stroller and it will be tagged
3. The baby can then be strolled till the plane`s gate (of course, they will ask for screeening, which is normal)..., .and you can hand over to crew member and you will receive it after each transit or destination.
4. A family travelling with infant is always given first priority to enter into the flight compared to other economic class traveller (they will announce anyway during boarding time).
5. While booking the ticket, you better tell the airlines to provide you all on the row close the bassinet, click here  for clear picture: 
6. http://4.bp.blogspot.com/_zqS5vfVsSsY/TVEfTZfFzpI/AAAAAAAAAzg/KG6aF9bBPKQ/s1600/bassinet2.jpg

I always make a phone call to the AIRLINES few weeks prior to the departure date just confirming what an infant can carry (usually, it is his  diaper bag, stroller,  10kg checked-in luggage, which depends on airlines).

bottom line, stroller will be only folded during flight hours, rest time - the lovely baby will be using it.... so mom and dad will have very easy time.  Since the baby is small, she or he may not require separte seat, mom/dad's lap is the best for the baby. You will be instructed by the crew member how to hold him during take off or landing time.

Best of luck.

PS. do not forget to carry at least 20 pcs of diaper and 2 packets of wipes.