So my daughter will have been on more planes before she is 18 months old than I had been on when I was 18 years old. We have learned a few tricks that make the flight easier. Of course every child's temperament is different and therefore every flight experience is different but here are some general tips to help.
First, try to pick you seats as close to the front as possible. They allows family's to board early but getting off the plane can be the cause of a lot of anxiety. After a long flight your cranky child just needs to get off and the closer to the front you are the quicker the process.
Don't worry to much about the other travelers around you. Usually if anyone is looking at us when we first board with the look that says "geez this is going to be a long flight" I usually apologise in advance and then try to keep her entertained. Most people are not bothered at all and have had similar experiences with their own children. Just be prepared with snacks, bottles or toys.
MOST IMPORTANT! After many flights I would be so frustrated when I would see empty seats and I asked the attendant before boarding if there was an extra seat for my child and they said no. I have learned a few tricks. Your best possible chance of actually getting the extra seat is booking yourself and spouse in the aisle and window seat. Leaving the middle open lowers the chances that someone will choose that seat. If the plane is full, obviously its not going to happen but in that case just politely ask the middle seat person if they would mind having the window. I have never met anyone who would rather sit wedged in between two people with their baby. They will probably be thanking you and also will be much happier they are sitting next to you and your adorable child. Now the trick here is that when you ask for an empty seat for you child, the attendant can only move an aisle seat to another aisle or a window to another window but a middle seat can be moved anywhere. That allows your chances of having the person in the seat next to you moved much higher! Of course if you are on Southwest its a free for all but on the other airlines this can mean the difference between space and an enjoyable flight or a cramped seat and many apologies.
Try to befriend the flight attendant. They can made your journey so much more enjoyable.