Hi D. -
I've traveled quite a bit with my three little ones, often traveling by myself with two at a time when they were babies and these tips worked for me.
1. If you can, try to plan your trip so that it coincides with a nap time while flying - not at security, at airport, etc.
2. Pack a bag full of activities that you know your child likes; i.e. favorite foods, books, toys. It and your diaper bag should be your only carry-ons so you are not too overloaded. If you are traveling with another person, you could bring one more but I wouldn't go beyond that.
3. If your child likes movies, you could always bring a portable DVD player.
4. Regarding the seat purchase, I have always bought a seat for my child and carried their car seat on. It was so much easier not to have to worry if there would be an available seat or to wrestle with an infant who doesn't want to sit on your lap. They get used to their car seat, so being strapped in it doesn't necessarily feel uncomfortable and it sounds like you are going to need it at the other end of your trip. Also, I once attended a Moms Group where a flight attendant came to speak and she said that in a case of real, severe turbulence you will never be able to hold on to a child. That pretty much did it for me. I thought that if anything ever happened to my child, I'd pay that few hundred dollars over and over again.
5. I also recommend having some Motrin or Tylenol handy in case their ears hurt. I always just gave them a dose an hour before flight just in case. It also helps some children to nap.
Hope this helps. Have a good time.
L.