M.S.
People reading this are gonna know me, cuz I like to answer these plane ride questions! LOL!
First of all, 99% of the time, the anxiety of the plane ride is worse than the actual ride. Prepare the best you can, then let it go:) You can only do so much.
Here are some things that have worked well for us over the years and we started flying when my 11 year old was 8 months. We've traveled with kids of every age so far and you'll have to figure out what will work best for your 3 year old.
Make sure you have an extra set of clothes for her (and you). Layer as much as possible. If there's a spill, hopefully, you can just pull off the outer layer and not have to actually change the shirt. If you get the spill, too, you'll not want to wear it the rest of the ride, either. Pack at least 1 or 2 more diapers than you think you'll need under normal circumstances. Yes, wipes are fine. But, since you can't take juice boxes or water bottles through security, you'll have to take an empty sippy cup and buy whatever drink you want for her after you get through security. THey'll be places to get drinks by your gate. A quart ziplock baggie will hold your hand sanitizer, Tylenol bottle, diaper cream-if you want, etc In addition to hand sanitizer, I took some CLorox wipes to wipe down the area around the kids seat. The amount of germs....ick. Some good snacks are baby carrots, crackers(in hard containers), anything that will not get mushy or crumble. Also, possibly a sucker or licorice stick for her to suck on (the swallowing is the key to get her ears to adjust) while the plane is taking off and landing. She can also drink while take-off, whatever gets her to swallow and equalize her ears.
The carseat is whatever you think she will be more comfortable in. We've done it both ways and if she won't keep taking her seatbelt off, then she may enjoy sitting in the seat by herself. If she's small, she may not be comfortable in the big seat and can't reach the tray very well. Do you have access to a booster seat or is it a 5 point harness? What will your options be if your carseat doesn't make it to your destination when checked?
Get your daughter some toys that are new-she hasn't seen them before. You could even wrap them like a birthday gift. Half the fun will be unwrapping them! Pull one out at a time to spread out the surprise. New books, crayons, markers, paper, coloring books, doll, etc.
We have also brought along a computer and had the kids watch a movie. We had three kids under three, so the movie was a great thing when they were all ages 2-5!
If you can fit it, take along a larger tupperware-type container. The lid, alone, also works. This will hold the cup, crayons, snacks, etc. so that they don't roll off the tray. It's no fun to try to retrieve these items from the floor under you. Whatever you can take that has raised edges to contain items. If you are ok with playdough and your daughter will give up some of it, I have made a "fence" around the edge of the tray to keep the markers on the tray. My daughter loved it, cuz it reminded her of a "horsey fence" and she used her little play horses inside the fence. THis is if you want to play with playdough during the ride.
Make sure your carry-on bag is easily accessible-something with a big main compartment that you can get to while it is under the seat in front of you. A small suitcase would not be the best choice. Sure, it can hold a lot of stuff, but it isn't something you can get into easily.
That's all I can think of for now. Have a great time and good luck!!