Hi T.,
I have taken my daughter on 3 airplane trips (all short flights): to Phoenix when she was 7 months old, and to Phoenix and Las Vegas this month (17 months old). They were all short flights, which I was very glad for. My daughter is too active, and I was by myself, so an hour on the plane is enough.
It's really helpful if you can get your side of the row to yourself. On the trips to Phoenix, we were on the little planes that have two rows of two seats. When she was 7 months old, I bought a seat for her and strapped her in her car seat. She took short naps on both flights. This time, I didn't buy her a seat on any of the flights, but on the Phoenix trip the men seated next to me moved "to give me more room" :-) and it was a huge help. The Las Vegas flight was a plane with two rows of three seats, and we had nobody else on our side on the way out there, and nobody seated in front of us. So it was nice because my daughter is obsessed with putting the tray table up and down, and kicked the seat a few times, but we didn't bother anybody. On the flight home, there was a lady on the aisle seat, but the middle seat was empty, and my daughter was so exhausted she slept the whole way - that was the best flight yet!! So my advice would be to buy a seat for her if you can afford it, or to travel during the non-busy times (Tuesday, Thursday, Sunday) to try to have as much space for yourself as you can. I booked a window seat on all the flights (if you bring the car seat on the plane it will need to go in the window seat), and my daughter liked looking at the window at the people on the ground, and at everything moving by as we took off.
Bring snacks, those are always good for keeping her occupied, and you'll need food if you get delayed. You can also bring a drink for your daughter - just separate it with the other liquids, and the TSA will have you open the bottle/cup so they can test it to make sure it's not an explosive (they won't put anything in the liquid, just wave this little strip thing over the top to pick up any fumes). I bring my daughter's favorite snacks: bananas, goldfish, animal crackers, grapes.
I have the Sit'N'Stroll carseat/stroller, which makes it very convenient for traveling and then you have a car seat at your destination. If you're renting a car, it might be easier to rent a car seat from the car rental place.
As far as keeping them occupied during the flight, good luck! I brought books and toys during the flight. My daughter did like the "Brown Bear" book on the first flight to Phoenix, but really wasn't interested in the books on our flights to Vegas. I bought the "Barrel of Monkeys" at Target, but that lasted for about 20 seconds until she started throwing them into the aisle and I took them away. I bought the Crayola Tadoodles at Target. My daughter isn't really interested in coloring, but she liked taking the crayons in and out of the holder. By the way, baby wipes take the crayon off of the airplane walls and tray with one swipe ;-) . If she has a favorite toy or book, bring that, and also bring something new that might hold her interest for a while. My daughter doesn't watch TV or movies yet, so I didn't even try that. We were on Jet Blue to and from Vegas, and she barely glanced at the TV screens.
The best thing would be if you can get her to sleep on the airplane, or for at least a portion of the flight. But any time I've tried to time her naps, I end up defeated!
Good luck to you!