Congratulations on your new little one!
At that age, my kids were still pretty easy on the plane. We would try to get seats where the engine noise and vibration were the loudest, and they would usually just fall asleep because the sound was soothing to them. The engine noise was also a plus because it would drown out some of the baby's noise if the baby got cranky, which rarely happened. I always brought a few new quiet toys--new toys usually hold interest for a few minutes at least!
We always bought a seat for baby, not just for safety but also because the baby would always sleep well in the carseat he was accustomed to. That's probably my best tip. And don't worry about the dirty looks--as long as you are staying calm yourself and doing your best to keep the baby happy, anyone who gives you a dirty look is just being rude and unkind. (I remember giving those "looks" before I had kids--HA!) I remember once when flying with my baby, as we landed I heard the man across the aisle complaining to the flight attendant that my baby had laughed too loud during the flight...which was probably true, since I had kept the baby giggling through the flight to keep him from crying. The flight attendant quietly apologized to the man and handed him a drink coupon for his next flight--I was mortified, but knew I had done my best and that was all I could do!
Have fun!