I've been flying with small children since my oldest was just a couple of months old, eight years ago. I can tell you that the first time I flew with her my anxiety was through the roof, and that was for a short, one hour flight. The next time we flew was to Europe when she was seven months old and it was just her and me, no support from my husband. I was a wreck going into the flight because
1. I just kept picturing her kicking & screaming the whole 11 hour flight
2. Her being awake and me having to keep her from crying the whole 11 hour flight
3. Me going absolutely crazy because I had no friggin clue what I was doing and I'm the type of person who cares way too much what other people think
Now, everything turned out pretty much ok except she ended up getting a fever towards the end of the flight...I never expected that to happen. And over the years, as I've added more kids to the mix and continued to travel, I've picked up on some tricks for flying with young children and infants that might help other new parents out. Keeping in mind that I like to travel as light as possible, my list is so you don't take unnecessary items that you won't end up using with you, which I've seen many parents do when traveling with children.
Also, strollers can be checked at the gate when boarding, car seats are recommended for children under two but lap seats are available, and milk & formula are able to pass through security without being dumped. However, I recommend bringing powdered formula just in case. Once a child is old enough for their own seat, they also get a carry-on bag *yay for an extra bag!
Traveling with small children and babies doesn't have to be a nightmare and even though unexpected events may occur during a long flight/journey (um, one time my daughter-14 months- and I got stranded at an airport after our 3rd flight of the day got cancelled at 11:00pm at night...on New Year's...after 15 hours of traveling. I seriously can't make this up). It's best to be prepared but without having to lug too many unnecessary items.
Good luck & safe travels!