This comes from my experience of United Airlines and several other random ones (Lufthansa, Delta). American Airlines is known for being especially bad about breaking chairs.

My Video on the Process from Picking a Place Through Booking Travel
While Booking Travel
Book with an airline less likely to break your chair. Here are the latest stats on breakage. American Airlines is the worst closely followed by Southwest. Allegiant was the best. Of the big airlines Delta is the best followed by United.
I recommend booking a seat as close to the front of the plane as you can to be closer to your seat. Ditto to the bathroom. In many airlines this means some form of “economy plus”, which is usually about a $50-$200 upgrade each way. This will help you get from the door to your seat. It doesn’t hurt to call the airline and see if they can put you in economy plus or equivalent without the charge based on disability. Every once in a while some nice person will do this for free.
This all depends on flying with an airline that allows you to chose your seat in advance. I highly recommend flying with one of those.
I have to travel in lay-down seats for flights longer than 3 hours. This means I can be right by the door they board in and near the restrooms. If you have a disability like mine where sitting up for a long time is impossible you may need to book the lay-down seats. If so, when you book look at the seat ma to confirm it is lay down seats, and then periodically check back on your booking to make sure they haven’t changed the plane.
Check policies about wheelchair travel for your airline. United has a page where you can see you what planes your chair will fit into. They also have an information form to help the baggage handlers. Others might have similar forms. Contact your airline.
Tell your airline ahead that you are traveling with a chair and what kind. Confirm that the flight is able to take it (some smaller planes can’t fit big wheelchairs).
Call or email TSA Cares ahead of your flight. They will meet you at the curb and get you through security.
Get a direct flight if you can Every stop is an opportunity for them to leave your chair behind or break it. The less stops the better.
If you have a connection, book a minimum of 2 hours between flights. This ensures that they have time to unload and re-load your chair, or if you are using your chair between flights, to unload your chair and get it to you, and then for you to get across the airport to your next check in before the beginning of boarding.
If you can avoid smaller regional planes, do. They have to hand-lift the chair in, and in some cases (I have learned from the baggage handlers) they store it sideways to make it fit. Both things mean it is way more likely to break.
Know Your Rights See the Air Carrier Access Act
What to Prepare Ahead
–Carry on Bag with all your essentials, all medical needs (including tolieting needs if that applies to you), toiletries and a change of clothing.
–Duffle Bag for Wheelchair Parts to hold things like your headrest, cushion (unless you are using it in the plane on your seat), leg rests if those detach, batteries if they are lithium, your charger, your cupholders, smaller pouches and bags attached to your chair, etc.
–Tag your Wheelchair with a Tile or AirTag. Also make sure it gets it’s destination tag at the gate.
Make sure your chair is in good shape– service it if it was not serviced recently. Otherwise tighten all the screws, etc.
Make sure you have a proper converter for your power chair: check what the power conversion needs to be for the places you are visiting. Your wheelchair charger may have a built-in converter that will work, but it may not. If it doesn’t and you don’t use a converter in addition to the plug adapter it could fry your wheelchair.
–Wear compression socks and comfortable clothing
-For Countries Outside the US bring a Proof of Disability with you: Your parking placard with its paperwork that has your name on it, an America the Beautiful Card, Veteran’s Services Card, etc.
-Obviously make sure your passport is valid for at least 3 months past your arrival in the country(s) you are visiting.
-RE Toileting: figure this out before you fly. If you cannot transfer easily to the plane’s toilet or walk a short distance you will need to take measures beforehand and have a strategy. This video might help.
Check in
Arrive early at the airport: 2-3 hours for domestic flights, 3-4 hours for international ones. Everything takes more time. Security can take a very long time to get your pat down, even when you are able to skip the line. And of course you board first, so being there at the beginning of boarding is important.
ALWAYS check in at the actual desk with an actual person, not a kiosk. This will ensure you can confirm your disability needs and information about your chair is all understood and correct.
I highly recommend checking it at the gate. You are entitled to do this. If you check it at the check-in desk they take your chair and give you a crappy one with a stranger pushing it. The experience is pretty dehumanizing and at the end they leave you sitting at the gate and take the chair. Need to go to the bathroom while you wait? Tough. Then another person with a chair will pick you up and take you to the plane during boarding.
If you check your chair for the gate you take your own chair through security and all the way to the entrance to the plane. Some airlines take your chair at the gate and take you the last bit to the plane in their chair.
Make sure they tag your chair and that you prepare it for a safe flight.
If you aren’t already seated near the front, ask at check in if you can moved closer to the front to make it easier to get to your seat. Sometimes they can do this. You can also ask again at the desk by the gate.
At check in and the gate they will also ask you if you need an “aisle chair”. If it is at all possible for you to walk to your seat using the chairs for stability or to rest, do that instead. Be familiar with the seat map so you have an idea of the distance you have to go. The aisle chair is this weird narrow chair and they strap you down so you can’t move and shove you to your seat.
If you need a aisle chair, the chances are you should also request an “onboard chair”. If this is the case for you you should see Cory Lee’s article here for more information relevant to your needs.
Going through Security
First, you usually get a priority line. Ask the airport personnel at the lines where to go. Once you bypass the regular line, you hang out waiting to go through the gate. This is because they are finding someone to pat you down. Since I am female and they need a female agent (less common) this can sometimes take a while. If you can stand up for long enough for the scanner and don’t mind sitting on the floor while they check your chair that’s an option. I did this once when they had no female agent on duty and there was no sign there would be one. I had already waited an hour and was going to miss my flight. This is probably rare because they are required to have a female agent at all times.
I advise keeping a travel companion close if you have one and having them watch the pat down. There is something about that job that brings out psychos and such people like to take advantage of people they see as helpless. Twice when alone I have had abusive agents literally ram their fingers into my lady parts through my clothes! They are supposed to use the side of their hand when they go between your legs, not shove the points of their fingers there. They are supposed to use the backs of their hands on your breasts, not grab and squeeze them. Backs of their hands on buttocks, not pounding fists. Be ready to advocate for yourself or have someone advocate for you or ask for a manager to observe. You have the right to have a manager present.
Make sure to let them know immediately about any medical devices to be careful of before they start. Make sure any bags or pouches attached to the chair are either empty or go through the scanner. They will also check your chair, which involves you leaning forward, and lifting one butt cheek at a time so they can look under you. Most times this whole process is totally fine, but when it isn’t it can be extremely violating.
Getting off the plane and getting your chair
Sit tight. Wait in your seat for everyone else to get off. Your chair won’t arrive for a while. You can work your way up towards the front and wait in a front seat of coach or business class and keep in contact with the flight attendants about your chair “I’m waiting for a wheelchair that looks like this, is it ready?”. The flight attendants should help you with your bags if you explain you are in a wheelchair.
If you checked your chair at the entrance of the plane they should bring it to the entrance of the plane. If they don’t ask the flight attendant to have them bring it. If they still don’t or if you checked your chair at the desk there will be a person with one of their chairs to take you to your own wheelchair. DO NOT LET THEM LEAVE YOU WITHOUT YOUR WHEELCHAIR, especially if you are traveling alone. If they can’t find it, make sure they stay with you until they do. They have a radio, etc. and can make it happen. You don’t want to be left in a random chair at the airport and forgotten about.
Be prepared to wait
I have waited 4 hours several times and over 5 hours once. If nothing goes wrong it will just be there at the gate for you. The better you prepare your chair and the notes on it, the less chances of things going wrong.
If they break your chair they have to repair it and give you a loaner for free. If you leave the airport they are no longer obligated. You have to ask the airline to fix the situation and file a report right away and not take no for an answer until everything is sorted out. You should also file a complaint with the department of Transportation here.
Leaving the Airport
make sure that before you take your trip you have identified a way to get your chair to where you are staying and around generally. Have a backup plan if your flight is delayed. Be prepared for what to do on the ground.
Good luck! I know it’s a little overwhelming, but the travel itself is totally worth it!
An article on this by Cory Lee, who cannot walk at all so has other important information I don’t have for those who cannot transfer by themselves.