Above all these are MY tips. Hope some can help you too.
We drive our car to the airport and use airportlax.com to book an indoor parking garage with security, just outside the airport to save hundreds of dollars (this time for example I paid $75 for 7 days, instead of $400 that would have been at the official airport). It's just 10min shuttle drive and the kids enjoy the experience. Tip when booking an outside airport parking: I look into indoor/outdoor option, security, price, shuttle frequency and distance. Upon landing so its right there when we get out of the terminal, so easy even more than walking to your car with the luggage in the airport parking.
We check in all of our bags so we are hands-free which allows us to move freely with the kids. We only keep 2 carry-ons and they are 2 backpacks, one for mom and one for dad, with all of the airport essentials: passports, healthier snacks, extra clothes/jackets, and activities for the kids.
I have a no stroller at airport rule since they can walk because: the more movement they get out in the airport, the less movement they need to release on the flight. Let's be realistic we simply can't expect them to sit down in both places!
I also make them do their own things at the airport, like go through customs with their own passport or through security with their own tray. This way they are excited and learn too!
Booking the right time of the flight is key for the whole experience, including the airport: -A flight of 2-6h I book during nap time -A flight of 7-8h+ I do overnight -A really short flight of 1h I would simply work around prioritizing that they can take a good nap elsewhere before or after - I try to never book a really early flight so we can all sleep a good night rest the night prior and avoid weaken our immune system and tantrums too
Food is key to minimize tantrums, get good naps and avoid over stimulation so I pack healthier snack options, buy some at the airport too, and avoid added sugars. - If the flight is long enough for them to eat on-board (great activity & distraction) bring extra food for the plane as good options tend to be limited, especially veggies, fruit and protein. - If they'll nap or sleep during the whole flight, make sure they get a good meal before boarding in a nicer restaurant.
I avoid screens and activities at the airport to save them for the flight (unless I know they'll sleep the whole flight)
Upon landing, have a car rental with car seats or car shuttle with car / seats lined up for pickup and transportation to destination.